ABSTRACT
Title of dissertation: CONTRIBUTIONS TO BAYESIAN
STATISTICAL MODELING
IN PUBLIC POLICY RESEARCH
Kevin D. Dayaratna, Doctor of Philosophy, 2014
Dissertation directed by: Professor Benjamin Kedem
Department of Mathematics
This dissertation improves existing Bayesian statistical methodologies and ap-
plies these improvements to a variety of important public policy questions. The
manuscript is divided into six chapters. The first chapter provides an overview
of the various chapters of the dissertation. The second chapter improves existing
Bayesian binary logistic regression methodologies using polynomial expansions as an
alternative to existing Markov Chain Monte Carlo (MCMC) methods. Our improve-
ments make the estimation technique quite useful for a variety of applications. We
also demonstrate the methodology to be considerably faster than existing MCMC
methods. These computational gains are quite useful for models analyzing large
data sets involving high-dimensional parameter spaces. We apply this methodology
to a child poverty data set to analyze the potential causes of child poverty. The next
chapter improves upon a well-known technique in semiparametric modeling known
as density ratio estimation. This methodology is useful in principle; however, it
suffers from one primary limitation - The technique has thus far been incapable
of modeling individual-level heterogeneity. Modeling heterogeneity is important as
there is often no a priori reason to believe that different individuals (or observations)
in a data set will behave in an identical manner. We ameliorate this limitation in
the third chapter of this dissertation by adapting density ratio estimation methods
to accommodate individual-level heterogeneity. We apply this new methodology to
an analysis of the efficacy of medical malpractice reform across the country. In the
fourth chapter of this dissertation, we shift our focus toward improving Bayesian
credible interval estimation via semiparametric density ratio estimation. We do so
by applying an innovative adaptation of the methodology, known as out of sample
fusion, to posterior samples from a hierarchical Bayesian linear model looking at
the efficacy of the welfare reform of the 1990s. In the fifth chapter, we extend the
application of this methodology to credible interval estimation of a hierarchical gen-
eralized linear model used for analyzing terrorism data in a number of major conflicts
across the globe. We use our results to offer some prescriptive policy suggestions
regarding counterterrorism policy. The final chapter concludes the dissertation and
offers a number of suggestions for further research. We emphasize that the modeling
contributions presented in this dissertation are useful in myriads of other applied
problems beyond just the public policy applications presented here.
CONTRIBUTIONS TO BAYESIAN STATISTICAL MODELING
IN PUBLIC POLICY RESEARCH
by
Kevin D. Dayaratna
Dissertation submitted to the Faculty of the Graduate School of the
University of Maryland, College Park in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
2014
Advisory Committee:
Professor Benjamin Kedem, Chair/Advisor
Professor Tobias Von Petersdorff
Professor Yuan Liao
Professor David Hamilton
Professor P.K. Kannan (Dean’s Representative)
c© Copyright by
Kevin D. Dayaratna
2014
Dedication
To Daddy
ii
Acknowledgments
My experience in Graduate School at the University of Maryland is one that
will remain with me for the rest of my life. There are so many people I’d like to ac-
knowledge that these first few pages cannot accommodate them all. Firstly, I would
like to thank both of my parents, Dr. Lama Dayaratna and Mrs. Esme Dayaratna, as
well as my two siblings, Arnal and Sandra, for their endless support and encourage-
ment over the course of my entire life. My accomplishments would almost surely not
have been possible without them. I’d also like to of course thank my wife Sashi for
her tremendous support as well. Others I’d also like to thank include Ron Matteotti
and my in-laws Mr. Gamini Abeysekara and Mrs. Kalyani Abeysekara. I’d also like
to thank many of my other relatives including Asantha Kempitya, Obashini Kempi-
tiya, Dharshika Kempitiya Sooriyabandara, Gihan Sooriyabandara, Hemali de Silva,
Harsha Rajapaksha, Udeshi Hargett, Nirmala Abeysinghe, and Channa Abeysinghe
to name a few.
There are also of course plenty of others outside my family whom I would like
to thank. Firstly, from the bottom of my heart, I would like to express my utmost
gratitude to my adviser Professor Benjamin Kedem. Professor Kedem has not only
been the best adviser I could have ever asked for but has also been a great friend and
colleague. Much of this dissertation builds upon many of his excellent ideas, and
he has provided tremendous input and guidance to improve upon previous drafts of
this dissertation.
I would also like to thank Professor Paul Smith who admitted me to the
iii
program and served as my professor for some of my coursework. In addition, I would
like to thank a number of other professors in the department including Professors
Eric Slud, Abram Kagan, Leonid Koralov, Sandra Cerrai, Tobias von Petersdorff,
Yuan Liao, and David Hamilton to name a few. Outside Maryland’s Mathematics
department, I would like to thank my colleague Professor P.K. Kannan of the Robert
H. Smith School of Business, whom I had the privilege of publishing other research
with outside this dissertation [22]. I would also like to thank Professor Jordan Boyd-
Graber of the iSchool for introducing me to the concept of non-parametric Bayesian
modeling used in Chapter 5 of this dissertation. Another faculty member I would
like to thank is Professor Cindy Clement of the Economics department for giving
me the opportunity to serve as a Lecturer in her department for each of the last
three years.
Additionally, there are numerous classmates at Maryland that I would also like
to thank including Eddie Kim, Xuan Yao, Xuan Liu, Joyce Hsiao, Kwame Okrah,
Hisham Talukder, Arseniy Zakharov, Joseph Paulson, David Shaw, Daniel Han,
Jong Jun Lee, Clare Wickman, Jim Greene, William Waldron, Catherine Ochalek,
Jeff Tan, Wenqing Hu, Wen Zhou, Lemeng Pan, Yue Tian, Zi Ding, Sean Burke, Joel
Witten, Peggy Tseng, Gauri Kulkarni, and Anshuman Sinha among others. I would
also like to especially thank Paul Tschirhart of the University’s Writing Fellows
Program for his advice about LaTeX coding and help formatting this dissertation. I
would also like to thank the Mathematics department’s great staff including Celeste
Regalado, Linette Berry, Haydee Hidalgo, and Alverda McCoy.
Outside the University of Maryland, there are also many people I would like to
iv
thank. Firstly, I am indebted to Professor Steven J. Miller of Williams College. Steve
has essentially served as my “mathematics mentor” since I was an undergraduate
at U.C. Berkeley. Over the years, we have become friends and colleagues, and I’ve
had the privilege of co-authoring a number of publications with him [23,24,108]. He
has also provided very insightful comments to this dissertation. I would also like to
thank Professor Eric Bradlow of the University of Pennsylvania for helpful input on
the Chapter 2 of this dissertation.
There are many people from my undergraduate days at the University of Cal-
ifornia, Berkeley I would also like to thank. Special thanks go to my good friend
Gagan Tara Nanda whom I studied mathematics with throughout the course of my
studies at Berkeley. I’d like to thank my undergraduate adviser, Professor L. Craig
Evans, who was instrumental in getting me interested in continuing to study higher-
level mathematics. I’d also like to of course thank a number of professors that I had
the privilege of learning from at Berkeley including Professors Richard Borcherds,
Maciej Zworski, Michael Hutchings, and John Neu. Studying mathematics from
these and other professors interested me greatly in pursuing graduate-level studies.
I would also like to thank one of my political science professors, Professor Darren
Zook, for getting me interested in terrorism analysis and ethnic conflict, which form
a component of this dissertation.
There are plenty of friends I’d also like to thank. I’d like to thank Alex
Levy who has been my good friend for nearly twenty years and the best man at
my wedding. I’d also like to thank Hovannes Abramyan, Mike Taylor, and Richard
Nazareth who not only served as groomsmen at my wedding but have also been great
v
friends since my time at U.C. Berkeley. I would especially like to thank Hovannes
and his colleague Mark Crain for providing the data that they used in their work with
the Pacific Research Institute for Chapter 3 of this dissertation. Other U.C. Berkeley
friends I would like to thank include Carl Densing, Elaine Gin, Yuriy Pasko, Mike
McFarlane, Dennis Bedford, Cameron Huey, Laura Russell, Nat Lipanovich, Erica
Batres, Erik Johannessen, Steve Berkovich, Michael Klein, Sam Lee, Eddie Kim,
Taiwan Udtamadilok, Rosanna Leroe-Munoz, David Timmons, and Rebekah Hsieh
among others. I’d also like to thank A.J. Hergenroder, Annie Moskovian, Jocelyn
Hsiao, Ben Ouyang, Laurie Kaufman, Rebekka Levy, Chris Luise, and Pamitha
Weeresinghe. Special thanks also go to the Rhodes, Pillai, and Pathirana families
for their friendship and encouragement over the years.
Finally, I would like to say thank you to many of my colleagues at The Her-
itage Foundation. In particular, I’d like to thank my colleagues in the Center for
Data Analysis - namely, Rea Hederman, Salim Furth, Drew Gonshorowski, Patrick
Tyrrell, James Sherk, David Kreutzer, David Muhlhausen, John Ligon, Rachel Gres-
zler, and Filip Jolevski. I’d also like to thank Derrick Morgan, Bill Beach, Robert
Rector, Leslie Merkle, Nina Owcharenko, Alyene Senger, Jennifer Marshall, Rachel
Sheffield, Bob Moffit, David Azerrad, Katie Tubb, Elinor Renner, and Pamela Ouzts.
I would especially like to thank Rea Hederman and Robert Rector for providing the
data used in Chapters 2 and 4 of this dissertation as well as for insightful discus-
sions. I would also like to think Jim Carafano for helpful comments on Chapter 5
of this dissertation on statistical inferences for counterterrorism policy.
vi
Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
List of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2. A Rigorous Examination of the Determinants of Child Poverty in America
via Closed-Form Bayesian Inferences with Implications for Welfare Reform 5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.1.1 What are the Causes of Child Poverty? . . . . . . . . . . . . . 5
2.1.2 Improvements to Bayesian Computation . . . . . . . . . . . . 6
2.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Polynomial Expansions of the Binary Logit Model . . . . . . . 10
2.2.2 Closed-Form Bayesian Inference via Polynomial Expansions
as Described In Miller et al (2006) . . . . . . . . . . . . . . . 13
2.2.3 Bayesian Inference via Series Expansions Using a Two-Sided
Heterogeneity Distribution . . . . . . . . . . . . . . . . . . . . 16
2.3 Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3.1 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 23
2.3.2 Comparisons to MCMC Methods . . . . . . . . . . . . . . . . 24
2.4 A Few Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.4.1 Incorporating covariances: Positing Heterogeneity from a Mul-
tivariate Normal Distribution . . . . . . . . . . . . . . . . . . 32
2.4.2 Allowing dependence on other factors . . . . . . . . . . . . . . 35
2.4.3 Arbitrary Priors . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 Application to Analyzing Potential Causes of Child Poverty . . . . . 38
2.5.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.5.2 Estimation Results . . . . . . . . . . . . . . . . . . . . . . . . 39
2.5.3 Policy Implications . . . . . . . . . . . . . . . . . . . . . . . . 41
2.6 Conclusions and Future Research . . . . . . . . . . . . . . . . . . . . 42
3. Closed-form Bayesian Inferences for Semiparametric Density Ratio Modeling
with Applications to Tort Reform . . . . . . . . . . . . . . . . . . . . . . . 45
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.1 Tort Reform . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.1.2 Bayesian Parametric methods . . . . . . . . . . . . . . . . . . 47
3.1.3 Density Ratio Estimation . . . . . . . . . . . . . . . . . . . . 47
3.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.2.1 Bayesian Density Ratio Estimation . . . . . . . . . . . . . . . 50
3.2.2 Advantages of Marginalization . . . . . . . . . . . . . . . . . . 52
3.2.3 Derivation of Distributions . . . . . . . . . . . . . . . . . . . . 52
3.3 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 A Few Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.4.1 Arbitrary Prior Distributions and Tilt Functions . . . . . . . . 58
3.4.2 Assuming homogeneity in the intercepts . . . . . . . . . . . . 60
3.5 An Application: Tort Reform . . . . . . . . . . . . . . . . . . . . . . 60
3.5.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5.2 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.5.3 Policy Implications . . . . . . . . . . . . . . . . . . . . . . . . 66
3.6 Conclusions and Future Research . . . . . . . . . . . . . . . . . . . . 67
4. Bayesian Inferences of Welfare Reform with Improved Credible Interval Es-
timation via Semiparametric Out of Sample Fusion . . . . . . . . . . . . . 69
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.1.1 “Ending Welfare as We Know it” . . . . . . . . . . . . . . . . 69
4.2 Statistically Modeling the Impact of Welfare Reform . . . . . . . . . 72
4.2.1 A Hierarchical Bayesian Model Allowing for State-Level Het-
erogeneity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.2.2 The Limitations of Bayesian Computational Methods . . . . . 74
4.2.3 Statistical Inferences Based on Density Ratio Estimation . . . 76
4.2.4 Empirical Likelihood Estimation in Semiparametric Density
Ratio Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.5 Numerical Simulations . . . . . . . . . . . . . . . . . . . . . . 81
4.3 The Impact of Welfare Reform . . . . . . . . . . . . . . . . . . . . . . 91
4.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
4.3.2 Analysis of the PROWA Using Bayesian Lagged Regression . . 94
4.3.3 Statistical Inferences Based on Hierarchical Bayesian Analysis
of PROWA . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
4.4 Additional Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
4.4.1 Policy Implications . . . . . . . . . . . . . . . . . . . . . . . . 106
4.5 Conclusions and Future Research . . . . . . . . . . . . . . . . . . . . 107
4.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
4.6.1 An Elementary Frequentist Analysis . . . . . . . . . . . . . . 108
5. Generalized Bayesian Inferences for Counterterrorism Policy with Improved
Credible Interval Estimation via Semiparametric Out of Sample Fusion . . 111
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
5.1.1 Combating Terrorism . . . . . . . . . . . . . . . . . . . . . . . 111
5.1.2 Dirichlet Process Priors and the Limitations Bayesian Para-
metric methods . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1.3 A Brief Review of Density Ratio Estimation . . . . . . . . . . 115
5.1.4 Using DRE to Improving on Bayesian Estimation Results . . . 116
viii
5.2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
5.3 A Bayesian Analysis of Several Major Conflicts Across the Globe . . 120
5.3.1 Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.3.2 The War in Afghanistan . . . . . . . . . . . . . . . . . . . . . 120
5.3.3 The War in Iraq . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.3.4 The Sri Lankan Civil War . . . . . . . . . . . . . . . . . . . . 127
5.3.5 The Troubles . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.3.6 The Use of Dirichlet Process Priors . . . . . . . . . . . . . . . 133
5.3.7 Improving Bayesian Credible Interval Estimates . . . . . . . . 135
5.4 Conclusions and Future Research . . . . . . . . . . . . . . . . . . . . 140
5.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.5.1 Variable Definition . . . . . . . . . . . . . . . . . . . . . . . . 144
6. Conclusions and Future Research - Where do we go from here? . . . . . . . 154
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List of Figures
3.1 Plot of Hˆ vs. H˜ - DRE, 2004 Per Capita Tort Loss Data . . . . . . . 64
3.2 Plot of Hˆ vs. H˜ - Bayesian DRE, 2004 Per Capita Tort Loss Data . . 64
3.3 Plot of Hˆ vs. H˜ - DRE, 2006 Per Capita Tort Loss Data . . . . . . . 65
3.4 Plot of Hˆ vs. H˜ - Bayesian DRE, 2006 Per Capita Tort Loss Data . . 65
4.1 Histogram of Posterior Sample for Posterior Intercept Mean Coeffi-
cient (µ0), TANF regression, Lag 1 . . . . . . . . . . . . . . . . . . . 95
4.2 Histogram of Posterior Slope Mean Coefficient (µ1), TANF regression,
Lag 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.3 Histogram of Posterior Intercept Variance Coefficient (σ20), TANF
regression, Lag 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
4.4 Posterior Slope Variance Coefficient, TANF regression (σ21), Lag 1 . . 97
5.1 Histogram of Posterior Sample for Suicide Attack Coefficient in Afghanistan,
Assuming a Normally Distributed Prior . . . . . . . . . . . . . . . . . 136
5.2 Histogram of Posterior Sample for Suicide Attack Coefficient in Afghanistan,
Assuming a DP prior . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
5.3 Histogram of Posterior Sample for Bombing/Explosion Attack Coef-
ficient in Iraq, Assuming a Normally Distributed Prior . . . . . . . . 138
5.4 Histogram of Posterior Sample for Bombing/Explosion Attack Coef-
ficient in Iraq, Assuming a DP prior . . . . . . . . . . . . . . . . . . . 139
5.5 Histogram of Posterior Sample for Facility/Infrastructure Coefficient
in Sri Lanka, Assuming a Normally Distributed Prior . . . . . . . . . 140
5.6 Histogram of Posterior Sample for Facility/Infrastructure Coefficient
in Sri Lanka, Assuming a DP prior . . . . . . . . . . . . . . . . . . . 141
5.7 Histogram of Posterior Sample for Hijacking Coefficient in North-
ern Ireland (before 1998 GF Agreement), Assuming a Normally Dis-
tributed Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
5.8 Histogram of Posterior Sample for Hijacking Coefficient in Northern
Ireland (before 1998 GF Agreement), Assuming a DP prior . . . . . . 143
5.9 Histogram of Posterior Sample for Unarmed Assault Coefficient in
Northern Ireland (after 1998 GF Agreement), Assuming a Normally
Distributed Prior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
5.10 Histogram of Posterior Sample for Unarmed Assault Coefficient in
Northern Ireland (after 1998 GF Agreement), Assuming a DP prior . 145
Chapter 1: Introduction
1.1 Introduction
In public policy research, questions abound about the direction of our country.
In recent years, rigorous statistical analysis has been a useful tool for answering many
of these questions [42]. This dissertation looks at a number of such policy questions
improving existing statistical methodologies in the process.
For example, in our next chapter, we look at the determinants of child poverty
and improve existing Bayesian estimation techniques to do so. Understanding the
root causes of poverty is a fundamental question that has consistently plagued policy
researchers for decades. With nearly $16 trillion spent on federal welfare programs
since President Lyndon Johnson began his War on Poverty and with the Obama
Administration expected to spend over another $10 trillion over the course of the
next decade, millions of American children continue to live in poverty [131]. Any
meaningful policy proposals should be based on addressing the causes of poverty
and not just the apparent symptoms.
Although a number of studies have attempted to identify these causes, no
studies, to our knowledge, have done so including individual-level heterogeneity in
the associated statistical models. Incorporating heterogeneity is very important as
different families will almost surely respond differently to potential causes. Unfor-
tunately, however, for the binary logistic regression models typically called upon to
help answer this question, positing a distribution for heterogeneity does not really
allow for closed-form inferences. As a result, researchers often have to resort to
numerical techniques such as Markov Chain Monte Carlo (MCMC) methods to es-
timate the associated models. These methods can be difficult and time consuming
to obtain convergence, particularly for large data sets.
We address this issue in Chapter 2 by presenting an alternative Bayesian es-
timation technique for binary logistic regression using polynomial expansions that
allows for closed form inferences, enabling researchers to make direct inferences
about the population. Miller, Bradlow, and Dayaratna (2006) also presented a sim-
ilar method; however, their result was quite limited as it was restricted to using a
single-sided prior distribution [108]. We assuage this limitation by allowing the re-
searcher to draw from one of the most flexible and commonly used prior distributions
- the normal distribution. After deriving our polynomial expansions, we present a
number of numerical simulations to illustrate the usefulness and advantages of our
approach. We also estimate our model on a large poverty data set from the Current
Population Survey to study the determinants of child poverty. In particular, we
find that marital status of parents, parental age, parental education, whether the
parents are working full time, and the number of children living in a household all
significantly influence whether a child grows up in poverty. We discuss the resulting
policy implications and conclude with a discussion of potential avenues for future
research.
In Chapter 3, we examine the efficacy of medical malpractice reforms instituted
throughout the country over the course of the last decade. In the process, we
improve on existing semiparametric density ratio estimation methodologies. Density
ratio estimation (DRE) is a well-known semiparametric modeling technique that has
been around for decades. Although the DRE method has proven to be very useful
2
in statistical modeling, it suffers from one primary limitation. In particular, the
method has thus far been incapable of modeling individual-level heterogeneity.
We ameliorate this limitation in Chapter 3 to enable DRE methods to model
individual-level heterogeneity. We perform a series of numerical simulations, along
with goodness of fit computations, to illustrate the efficacy of our approach. We
show that this new approach outperforms existing semiparametric density ratio esti-
mation methods. After our numerical simulations, we apply our approach to medical
malpractice loss data from the previous decade to quantify the probability of ex-
treme losses. Our results indicate the success of some recently instituted medical
malpractice reforms.
Subsequently, in our fourth chapter, we shift our focus toward welfare policy,
conducting a rigorous Bayesian analysis of the welfare reform of the 1990s, providing
some improvements to Bayesian credible interval estimation techniques in the pro-
cess. First popularized by Ronald Reagan in his classic 1964 “A Time for Choosing”
speech, the concept of welfare reform has been a hot topic of public policy research
for decades [127]. The Personal Responsibility and Work Authorization Act of 1996
was one of the nation’s most comprehensive efforts at welfare reform [118]. The law’s
primary aim was to transform one of America’s major welfare programs away from
a system fostering dependency and into a program providing temporary assistance
to enable people to become contributing members of society.
In Chapter 4, we utilize hierarchical Bayesian linear modeling to rigorously
quantify the impact of this law. In the process, we improve upon existing Bayesian
interval estimation methods by calling upon semiparametric DRE method thus far
used only for frequentist statistical modeling. We find that the welfare reform of
the 1990s was quite successful in getting people back to work and can be improved
upon even further. We conclude by discussing the resulting policy implications.
In Chapter 5, we shift our focus to foreign policy, particularly toward combat-
3
ing terrorism. Terrorism has been around for generations and understanding how
to fight terrorists has been a question vexing policymakers throughout the world.
In Chapter 5, we rigorously analyze terrorist attack data from four major conflicts
across the globe - from Afghanistan, Iraq, Sri Lanka, and Northern Ireland - to
help policymakers answer this very question. We utilize both parametric and non-
parametric (hence generalized) Bayesian logistic regression techniques to do so, and
again improve upon Bayesian credible interval estimation in the process by using
semiparametric DRE methods. We thus extend the application of the semipara-
metric DRE method used in Chapter 4 for linear models to this generalized linear
model. Our study helps shed light on the factors that influence the success of terror-
ist attacks in each conflict, providing policymakers with advice about how to more
effectively combat this very dangerous enemy.
Finally, in Chapter 6, we discuss some conclusions as well as potential avenues
for future research. We emphasize that the modeling contributions presented here
are applicable to myriads of other fields beyond just the public policy applications
looked at in this dissertation.
4
Chapter 2: A Rigorous Examination of the Determinants of Child
Poverty in America via Closed-Form Bayesian Inferences
with Implications for Welfare Reform
2.1 Introduction
2.1.1 What are the Causes of Child Poverty?
Ever since President Lyndon Johnson began his War on Poverty in 1964, the
U.S. Federal Government has spent an exorbitant amount (estimated to be as much
as $15.9 trillion) on welfare programs and on other forms of cash assistance to the
poor. Such programs include means-tested welfare programs such as Temporary
Assistance for Needy Families (TANF), the Earned Income Tax Credit (EITC),
Supplemental Security Income (SSI), food stamps, public housing, and Medicaid
among others [131]. As millions of Americans continue to live in poverty today,
some researchers have argued that many of these programs have been largely inef-
fective and have the potential to trap people in poverty. In his 1984 seminal work
Losing Ground, for example, Charles Murray of the American Enterprise Institute
suggested that many of these welfare programs were actually encouraging depen-
dence and ironically crippling the people they were intended to help [111]. Many
such researchers have consequently argued that meaningful public policy proposals
on these issues should be grounded on an understanding of the root causes of poverty
and not just the mere symptoms [130,136].
Although some studies have attempted to identify these causes, no such re-
search, to our knowledge, have incorporated individual-level heterogeneity in its
statistical modeling [130,145]. Incorporating heterogeneity is of utmost importance
in these models as there is no a priori reason to believe that all families will respond
to potential causes in the same manner. From a statistical perspective, assuming
homogeneous response coefficients in a model can lead to the researcher ignoring
potential variability and can consequently result in misleading statistical inferences.
We fill this gap in the extant literature by examining the potential causes of
child poverty via Bayesian logistic regression. The contribution of our study is two
fold. From a policy perspective, we look at the determinants of child poverty, while
incorporating individual-level heterogeneity into our model. Our incorporation of
individual-level heterogeneity allows us to make more informative statistical infer-
ences compared to methods that simply just assume homogeneity. Methodologically,
our approach makes Bayesian estimation involving large data sets, such as the child
poverty data set examined here, much more feasible.
2.1.2 Improvements to Bayesian Computation
From fields ranging from economics to public policy to medicine to professional
sports, logistic regression has become one of the most widely used tools in applied
statistical research. For example, logistic regression models have been used to help
understand the determinants of depression and suicide, examine consumer choice
in marketing research, model medical outcomes pertaining to various illnesses, and
shed light on baseball player hitting performance among others [1, 7, 11, 58–60, 133,
164,167].
With improvements in statistical computing power over the course of the past
6
two decades, incorporating heterogeneity in these and other models has become
increasingly common in the applied statistical literature. Researchers have incor-
porated heterogeneity by choosing amongst a variety of methodologies, including
a parametric Bayesian approach, a non-parametric Bayesian approach, or even a
frequentist finite mixture approach [49, 72, 74, 100]. Parametric Bayesian models
are often used to incorporate individual-level heterogeneity. Generally, with limited
data per individual in a data set, assuming a different parametrization for each
individual renders a model statistically unidentifiable, making estimation virtually
impossible. Researchers will typically assume that these individual response coeffi-
cients are all drawn from their own lower-dimensional probability distribution. They
can then estimate these models from an empirical Bayesian perspective or from a
fully Bayesian perspective by imposing priors on the parameters of the heterogeneity
distributions themselves [49,110].
Although “nice” in principle, incorporating individual-level heterogeneity is
often concomitant with the drawbacks of the computational complexity associated
with numerical computation. For example, numerical methods such as quadrature,
simulated maximum likelihood, and Markov Chain Monte Carlo (MCMC) methods
can be difficult and time consuming to estimate, especially for large data sets in-
volving high-dimensional parameter spaces [50, 138]. Additionally, commonly-used
MCMC methods suffer from the drawback of sensitivity to starting values and can
consequently result in a significant amount of simulation error.
As a result, a number of researchers have approached alternative techniques
to make Bayesian computation more feasible. For example, Everson and Bradlow
(2002) used polynomial expansions to approximate the posterior distributions of
the beta-binomial random variables using a class of prior distributions previously
considered non-conjugate [39]. Similarly, Bradlow et al (2002) used polynomial
expansions to improve on researchers’ ability to make posterior inferences about the
7
negative binomial distribution [9]. In subsequent research, McShane et al (2008)
used similar techniques to improve on Weibull count model estimation [105].
Miller et al (2006) used polynomial expansions to solve the problem looked at
in this research, namely for binary logistic regression [108]. Their approach, however,
suffered from a serious limitation by requiring that the prior distribution be single-
sided. Consequently, their result, although nice in principle, is very limited in scope
as it is often difficult to know a priori the signs of the coefficients beforehand. We
assuage this limitation by looking at the same problem but instead allowing the
researcher to draw from the one of the richest and most commonly used two-sided
prior distributions - The normal distribution.
In particular, we derive a marginalized likelihood for the binary logit model us-
ing polynomial expansions. We begin by reviewing the Miller et al (2006) approach
and its subsequent limitations. We then proceed by ameliorating these limitations,
assuming that the response coefficients are drawn from independent normal distri-
butions. We subsequently generalize this result by allowing correlations amongst
the coefficients. Afterwards, we allow for dependence of the prior distributions on
various covariates and then subsequently allow for the choice of non-normal prior
distributions.
Our model can be estimated via the method of maximum marginal likelihood
(MML) from which empirical Bayesian inferences can be made, allowing us to make
direct inferences about the population [110,147]. We present a number of numerical
simulations to illustrate the usefulness of our approach as well as its advantages
over existing MCMC methods. We also utilize our approach to answer an impor-
tant question in public policy research; particularly, identifying the causes of child
poverty.
8
2.2 Problem Formulation
Consider a data set obtained from i ∈ {1, . . . , I} individuals (units) having j ∈
{1, . . . , J} categories (e.g. illness, product brand, etc) measured on t ∈ {1, . . . , Ni}
occasions (repeated measures). As is standard, we define
yijt =



1 if outcome occurs for individual i pertaining to category j at time t
0 otherwise,
(2.1)
where pijt = Prob(yijt = 1) is the probability of a particular outcome occurring (e.g.
living in poverty, purchase of a product, mortality of a patient) for the ith individual
pertaining to the jth category on the tth occasion. Additionally, let p = 1, . . . , P
represent a set of attributes pertaining to the covariates, with corresponding values
xijt,p ≥ 0 such that XTijt = (xijt,1, . . . , xijt,P ). To account for residual effects not
manifested in the coefficient estimates for the explanatory variables, we can allow
xijt,1 = 1 defining category-level intercepts.
Multiplying over all individuals, categories, and occasions, we obtain the stan-
dard logit likelihood of the data, Y = (yijt):
P (Y |β) =
I∏
i=1
J∏
j=1
Ni∏
t=1
eyijtX
T
ijtβi
1 + eX
T
ijtβi
, (2.2)
where βi = (βi,1, . . . , βi,P ) is the coefficient vector for the ith individual with pth
variable-specific coefficient, βi,p and β = (β1, . . . , βI).
Allowing our model to accommodate heterogeneity is quite important in mod-
eling real world phenomena, as there is no reason to believe that all individuals will
behave in an identical manner. For example, in modeling consumer purchase behav-
ior, customers purchasing a product will almost surely differ in how they respond to
9
prices or promotions. In modeling baseball hitting performance, different sluggers
will respond differently to different pitching styles. Additionally, in public policy
research, looked at in this study, different families will respond differently to various
factors that may or may not contribute to them living in poverty.
We can model heterogeneity across individuals by allowing each βi,p to be
drawn from probability distributions. To start out, we discuss the Miller et al
(2006) approach and the limitations of their result. Subsequently, we ameliorate
these limitations and make the mode much for useful for applying to real-world
problems.
2.2.1 Polynomial Expansions of the Binary Logit Model
We are interested in the following marginalized likelihood which we intend to
maximize over our parameter space:
P (Y |Ω) =
∫
β
P (Y |β)N(β|Ω)dβ. (2.3)
In the above equation, P (Y |β) is our standard logit likelihood with a prior dis-
tribution N(β|Ω) and Ω represents the parameters of our prior distribution. As men-
tioned above, we have non-negative explanatory variables XTijt = (xijt,1, . . . , xijt,P )
and binary dependent variables Y = (yijt). We intend to maximize the above
marginalized likelihood over our prior distribution’s parameter space. Specifically,
our logit likelihood is:
P (Y |β) =
I∏
i=1
J∏
j=1
Ni∏
t=1
eyijtX
T
ijtβi
1 + eX
T
ijtβi
, (2.4)
It is the heterogeneity across i = 1, ..., I individuals in their βi,p coefficients that
we are modeling by allowing these parameters to follow N(β|Ω). Due to the fact
that the βi appears in both the numerator and denominator of (2.4), performing
10
the integration in (2.3) analytically for most choices of heterogeneity distributions
without any numerical approximations is essentially impossible.
We can take a series expansion approach to this problem and rewrite P (Y |β)
as follows:
P (Y |β) =
I∏
i=1
J∏
j=1
Ni∏
t=1
eyijtX
T
ijtβi
1 + eX
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
eyijtX
T
ijtβi
I∏
i=1
J∏
j=1
Ni∏
t=1
1
1 + eX
T
ijtβi
P (Y |β) = P1(Y |β)P2(Y |β). (2.5)
We refer to the second factor above as P2(Y |β) although it does not depend on
Y. If we assume XTijtβi < 0, we can expand P2(Y |β) via a geometric series expansion
as follows [108]:
P2(Y |β) =
I∏
i=1
J∏
j=1
Ni∏
t=1
1
1 + eX
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
∞∑
kijt=0
(−1)kijtekijtX
T
ijtβi . (2.6)
Putting together the pieces, we therefore have when XTijtβi < 0:
11
P (Y |β) =
I∏
i=1
J∏
j=1
Ni∏
t=1
eyijtX
T
ijtβi
1 + eX
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
eyijtX
T
ijtβi
I∏
i=1
J∏
j=1
Ni∏
t=1
1
1 + eX
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
eyijtX
T
ijtβi
I∏
i=1
J∏
j=1
Ni∏
t=1
∞∑
kijt=0
(−1)kijtekijtX
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
∞∑
kijt=0
(−1)kijteyijtX
T
ijtβi+kijtX
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
∞∑
kijt=0
(−1)kijte(yijt+kijt)X
T
ijtβi
P (Y |β) =
I∏
i=1
J∏
j=1
Ni∏
t=1
∞∑
kijt=0
(−1)kijte(yijt+kijt)
∑P
p=1 xijt,pβi,p . (2.7)
If, on the other hand, we assume XTijtβi > 0, we can also use a geometric series
expansion:
P2(Y |β) =
I∏
i=1
J∏
j=1
Ni∏
t=1
1
1 + eX
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
e−X
T
ijtβi
1 + e−X
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
e−X
T
ijtβi
1
1 + e−X
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
e−X
T
ijtβi
I∏
i=1
J∏
j=1
Ni∏
t=1
∞∑
kijt=0
(−1)kijte−kijtX
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
∞∑
kijt=0
(−1)kijte−X
T
ijtβi−kijtX
T
ijtβi
P2(Y |β) =
I∏
i=1
J∏
j=1
Ni∏
t=1
∞∑
kijt=0
(−1)kijte−(1+kijt)
∑P
p=1 xijt,pβi,p . (2.8)
12
And again putting together the pieces:
P (Y |β) =
I∏
i=1
J∏
j=1
Ni∏
t=1
eyijtX
T
ijtβi
1 + eX
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
eyijtX
T
ijtβi
I∏
i=1
J∏
j=1
Ni∏
t=1
1
1 + eX
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
eyijtX
T
ijtβi
I∏
i=1
J∏
j=1
Ni∏
t=1
∞∑
kijt=0
(−1)kijte−(1+kijt)X
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
∞∑
kijt=0
(−1)kijteyijtX
T
ijtβi−(1+kijt)X
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
∞∑
kijt=0
(−1)kijte(yijt−1−kijt)X
T
ijtβi
P (Y |β) =
I∏
i=1
J∏
j=1
Ni∏
t=1
∞∑
kijt=0
(−1)kijte(yijt−1−kijt)
∑P
p=1 xijt,pβi,p . (2.9)
In the next section we utilize these series expansions to derive closed-form ex-
pressions from which we can make Bayesian inferences.
2.2.2 Closed-Form Bayesian Inference via Polynomial Expansions as Described In
Miller et al (2006)
Ideally, one would like to allow each βi,p to follow two sided prior distribu-
tion, under such circumstances we would have a combination of both of the above
situations, as well as when XTijtβi = 0:
13
P (Y |β) =
I∏
i=1
J∏
j=1
Ni∏
t=1
eyijtX
T
ijtβi
1 + eX
T
ijtβi
=
I∏
i=1
J∏
j=1
Ni∏
t=1
[
eyijtX
T
ijtβi
1 + eX
T
ijtβi
(1)
]
=
I∏
i=1
J∏
j=1
Ni∏
t=1
[
eyijtX
T
ijtβi
1 + eX
T
ijtβi
[
I(XTijtβi > 0) + I(X
T
ijtβi < 0) + I(X
T
ijtβi = 0)
]
]
P (Y |β) =
I∏
i=1
J∏
j=1
Ni∏
t=1
[
eyijtX
T
ijtβi
1 + eX
T
ijtβi
I(XTijtβi > 0) +
eyijtX
T
ijtβi
1 + eX
T
ijtβi
I(XTijtβi < 0)
+
eyijtX
T
ijtβi
1 + eX
T
ijtβi
I(XTijtβi = 0)
]
.
This can be rewritten as:
P (Y |β) =
I∏
i=1
J∏
j=1
Ni∏
t=1


∞∑
kijt=0
(−1)kijte(yijt−1−kijt)
∑P
p=1 xijt,pβi,pI(
P∑
p=1
xijt,pβi,p > 0)
+
∞∑
kijt=0
(−1)kijte(yijt+kijt)
∑P
p=1 xijt,pβi,pI(
P∑
p=1
xijt,pβi,p < 0)
+
eyijt
∑P
p=1 xijt,pβi,p
1 + e
∑P
p=1 xijt,pβi,p
I(
P∑
p=1
xijt,pβi,p = 0)
]
. (2.10)
As a result, the marginalized likelihood is:
P (Y |Ω) =
∫
β
P (Y |β)N(β|Ω)dβ
=
∫
β
I∏
i=1
J∏
j=1
Ni∏
t=1
eyijtX
T
ijtβi
1 + eX
T
ijtβi
N(β|Ω)dβ
=
∫
β
I∏
i=1
J∏
j=1
Ni∏
t=1
eyijtX
T
ijtβi
1 + eX
T
ijtβi
dP βi,p
P (Y |Ω) =
∫
β
P (Y |β)dP βi,p
14
where Pβi,p is the measure induced by βi,p on measurable space (Si,p, Fi,p).
When Miller et al (2006) looked at this problem, the authors attempted to
integrate each βi,p individually for every potential value of i and p [108]. For a two-
sided heterogeneity distribution, such as a normal heterogeneity distribution, this
marginalization would involve:
P (Y |Ω) =
∫
β
P (Y |β)N(β|Ω)dβ
=
∫
β
I∏
i=1
J∏
j=1
Ni∏
t=1


∞∑
kijt=0
(−1)kijte(yijt−1−kijt)
∑P
p=1 xijt,pβi,pI(
P∑
p=1
xijt,pβi,p > 0)
+
∞∑
kijt=0
(−1)kijte(yijt+kijt)
∑P
p=1 xijt,pβi,pI(
P∑
p=1
xijt,pβi,p < 0)
+
eyijt
∑P
p=1 xijt,pβi,p
1 + e
∑P
p=1 xijt,pβi,p
I(
P∑
p=1
xijt,pβi,p = 0)
]
·
P∏
p=1
1
√
2piσp
· e
−(βi,p−µp)
2
2σ2p dβi,p.(2.11)
As the range of βi,p is the entire real line, the limits of the integration space
differ for the first and second integrals depending on whether
∑P
p=1 xijt,pβi,p < 0 or
∑P
p=1 xijt,pβi,p > 0. Miller et al (2006) noted that integrating over both spaces
would result in “numerous, complicated subdivisions of the integration space.”
These subdivisions, they argued, rendered the integration “untenable” and pre-
cluded the derivation of “tractable closed-form expansions.” As a result, the au-
thors restricted their model to adhere to only one of the above cases, particularly
XTijtβi < 0
1 and required that the density N(β|Ω) being integrated over be a one-
sided probability distribution. Making the assumption that N(β|Ω) was composed
of independent gamma distributions g(βi,p|bp, np) with parameters bp and np (i.e.
N(β|Ω) =
∏P
p=1 g(βi,p|bp, np)), they derived the marginalized likelihood as follows:
1 Miller et al (2006) actually parametrized their logit likelihood in a slightly different functional
form as P (Y |β) =
∏I
i=1
∏J
j=1
∏Ni
t=1
e
−yijtX
T
ijtβi
1+e
−XTijtβi
and hence equivalently assumed that XTijtβi > 0
and βi,p ≥ 0 ∀ i, p.
15
P (Y |Ω) =
∫
β
P (Y |β)N(β|Ω)dβ
=
∫
β
I∏
i=1
J∏
j=1
Ni∏
t=1


∞∑
kijt=0
(−1)kijte(yijt+kijt)
∑P
p=1 xijt,pβi,p
P∏
p=1
g(βi,p|bp, np)dβi,p


=
I∏
i=1
∞∑
ki11=0
· · ·
∞∑
kiJNi=0
(−1)
∑J
j=1
∑Ni
t=1 kijt
P∏
p=1
∫ ∞
βi,p=0
e−
∑J
j=1
∑Ni
t=1(yijt+kijt)xijt,pβi,p
·
1
bpΓ(np)
(
βi,p
bp
)np−1
e−βi,p/bpdβi,p
=
I∏
i=1
∞∑
ki11=0
· · ·
∞∑
kiJNi=0
(−1)
∑J
j=1
∑Ni
t=1 kijt
P∏
p=1
(
1
1 + bp
∑J
j=1
∑Ni
t=1(yijt + kijt)xijt,p
)np
Having made assumptions that the explanatory variables Xijt,p were restricted
to the set of non-negative integers, Miller et al (2006) borrowed some tools from an-
alytic number theory to rewrite the above equation in terms of solutions to a system
of Diophantine equations, which made estimating the model significantly more fea-
sible from a computational perspective [109]. The interested reader is referred to
Miller et al (2006) for a complete discussion of this methodology [108].
2.2.3 Bayesian Inference via Series Expansions Using a Two-Sided Heterogeneity
Distribution
Although the Miller et al (2006) result is elegant mathematically, it is not
particularly useful to implement in practice as in most applications it is generally
unrealistic to a priori assume that the regression coefficients all have the same sign.
However, for the case when J = 1 and Ni = 1, a simple transformation of variables
leads to very clean and tractable integration, allowing us to integrate within distinct
regions along the real line. Restricting J and Ni in this manner is quite reasonable
for many applied statistical problems including cross-sectional data analysis with a
single category (such as the child poverty application looked at later in this study),
longitudinal analysis of a single individual (such as the baseball player hitting streak
16
analysis conducted in Albrght (1993)), or analysis where the heterogeneity can be
assumed across all observations of the data set (such as the terrorist attack data
analysis conducted in Kyung et al (2012) or the data set used in the analysis of
medical outcomes in Wisner (1990)) [1, 90,108,164].
Specifically, if we make the assumption that pi = Prob(yi = 1) is the prob-
ability of a particular outcome occurring (e.g. living in poverty, patient mortality,
purchase incidence, etc) for the ith individual and again let p = 1, . . . , P represent a
set of attributes describing the covariates, with corresponding values xi,p such that
XTi = (xi,1, . . . , xi,P ) and take the product across all individuals i, the likelihood
function is:
P (Y |β) =
I∏
i=1
eyiX
T
i βi
1 + eX
T
i βi
, (2.12)
where βi = (βi,1, . . . , βi,P ) and β are defined as before. Upon making these assump-
tions, we can recall the marginalization presented in (2.11):
P (Y |Ω) =
∫
β
P (Y |β)N(β|Ω)dβ
=
∫
β
I∏
i=1
[
∞∑
ki=0
(−1)kie(yi−1−ki)
∑P
p=1 xi,pβi,pI(
P∑
p=1
xi,pβi,p > 0)
+
∞∑
ki=0
(−1)kie(yi+ki)
∑P
p=1 xi,pβi,pI(
P∑
p=1
xi,pβi,p < 0) +
eyi
∑P
p=1 xi,pβi,p
1 + e
∑P
p=1 xi,pβi,p
I(
P∑
p=1
xi,pβi,p = 0)
]
·
P∏
p=1
1
√
2piσp
e
−(βi,p−µp)
2
2σ2p dβi,p. (2.13)
In particular, since we are assuming that the βi,p follow independent nor-
mal distributions for p = 1, ..., P (i.e. βi,p ∼ N(µp, σ2p)) it follows that zi =
∑P
p=1 xi,pβi,p ∼ N(
∑P
p=1 xi,pµp,
∑P
p=1 x
2
i,pσ
2
p). Therefore, if we define Pzi as the mea-
sure induced by zi on measurable space (Ti, Gi) having density with respect to
Lebesgue measure:
17
f(zi) =
1
√
2pi
∑P
p=1 x
2
i,pσ2p
e
−(zi−
∑P
p=1 xi,pβi,p)
2
2
∑P
p=1 x
2
i,pσ
2
p ,
then:
P (Y |Ω) =
∫
β
P (Y |β)N(β|Ω)dβ
=
∫
β
I∏
i=1
eyiX
T
i βi
1 + eX
T
i βi
N(β|Ω)dβ
=
∫
β
I∏
i=1
eyiX
T
i βi
1 + eX
T
i βi
dP βi,p
P (Y |Ω) =
∫
β
P (Y |β)dP βi,p .
Applying our transformation we can see that [149]:
P (Y |Ω) =
∫
β
P (Y |β)dP βi,p
=
∫
TI
...
∫
T1
I∏
i=1
P (yi|zi)dP zi
=
I∏
i=1
∫
Ti
P (yi|zi)dP zi
=
I∏
i=1
∫
zi
P (yi|zi)
1
√
2pi
∑P
p=1 x
2
i,pσ2p
e
−(zi−
∑P
p=1 xi,pµp)
2
2
∑P
p=1 x
2
i,pσ
2
p dzi
P (Y |Ω) =
I∏
i=1
Hi, (2.14)
where:
18
Hi =
∫ ∞
−∞
[
∞∑
ki=0
(−1)kie(yi−1−ki)ziI(zi > 0)
+
∞∑
ki=0
(−1)kie(yi+ki)ziI(zi < 0) +
eyizi
1 + ezi
I(zi = 0)
]
·
1
√
2pi
∑P
p=1 x
2
i,pσ2p
e
−(zi−
∑P
p=1 xi,pµp)
2
2
∑P
p=1 x
2
i,pσ
2
p dzi.
We can decompose Hi into a sum of three integrals, Hi,1, Hi,2, and Hi,3 where
Hi = Hi,1 +Hi,2 +Hi,3.
Before we proceed, we present a simple integration lemma for integrating an ex-
ponential against a normal distribution with mean µ and variance σ2.
Lemma 2.2.1 (Integrating an Exponential Against a Normal Distribution).
∫ ∞
c
ekx
1
√
2piσ
e
−(x−µ)2
2σ2 dx = ekµ+
k2σ2
2 Φ
(
kσ2 − c+ µ
σ
)
(2.15)
∫ c
−∞
ekx
1
√
2piσ
e
−(x−µ)2
2σ2 dx = ekµ+
k2σ2
2 Φ
(
−
kσ2 − c+ µ
σ
)
(2.16)
where Φ(x) is the normal cumulative distribution function.
Proof:
19
∫ ∞
c
ekx
1
√
2piσ
e
−(x−µ)2
2σ2 dx =
∫ ∞
c
1
√
2piσ
ekx−
(x−µ)2
2σ2 dx
=
∫ ∞
c
1
√
2piσ
e
−1
2σ2
(x−µ)2+kxdx
=
∫ ∞
c
1
√
2piσ
e
−1
2σ2
[[x−(σ2k+µ)]2−(σ2k+µ)2+µ2]dx
=
1
√
2piσ
e
−1
2σ2
[−(σ2k+µ)2+µ2]
∫ ∞
c
e
−1
2σ2
[x−(σ2k+µ)]2dx
= e
σ2k2
2 +kµ
[
1− Φ
(
c− (σ2k + µ)
σ
)]
∫ ∞
c
ekx
1
√
2piσ
e
−(x−µ)2
2σ2 dx = ekµ+
k2σ2
2 Φ
(
kσ2 − c+ µ
σ
)
. (2.17)
The computation of the second integral is quite similar. As a result of (Lemma
2.2.1),
Hi,1 =
∫ ∞
0
∞∑
ki=0
(−1)kie(yi−1−ki)zi
1
√
2pi
∑P
p=1 x
2
i,pσ2p
e
−(zi−
∑P
p=1 xi,pµp)
2
2
∑P
p=1 x
2
i,pσ
2
p dzi
=
1
√
2pi
∑P
p=1 x
2
i,pσ2p
∞∑
ki=0
(−1)ki
∫ ∞
0
e(yi−1−ki)zie
−(zi−
∑P
p=1 xi,pµp)
2
2
∑P
p=1 x
2
i,pσ
2
p dzi
Hi,1 =
∞∑
ki=0
(−1)kie
(yi−1−ki)
2∑P
p=1 x
2
i,pσ
2
p
2 +(yi−1−ki)
∑P
p=1 xi,pµp
·Φ


(yi − 1− ki)
∑P
p=1 x
2
i,pσ
2
p +
∑P
p=1 xi,pµp
√∑P
p=1 x
2
i,pσ2p

 .
(2.18)
20
Hi,2 =
∫ 0
−∞
∞∑
ki=0
(−1)kie(yi+ki)zi
1
√
2pi
∑P
p=1 x
2
i,pσ2p
e
−(zi−
∑P
p=1 xi,pµp)
2
2
∑P
p=1 x
2
i,pσ
2
p dzi
=
1
√
2pi
∑P
p=1 x
2
i,pσ2p
∞∑
ki=0
(−1)ki
∫ 0
−∞
e(yi+ki)zie
−(zi−
∑P
p=1 xi,pµp)
2
2
∑P
p=1 x
2
i,pσ
2
p dzi
Hi,2 =
∞∑
ki=0
(−1)kie
((yi+ki))
2∑P
p=1 x
2
i,pσ
2
p
2 +(yi+ki)
∑P
p=1 xi,pµp
·Φ

−
(yi + ki)
∑P
p=1 x
2
i,pσ
2
p +
∑P
p=1 xi,pµp
√∑P
p=1 x
2
i,pσ2p

 .
(2.19)
Hi,3 = 0 as it is an integral against a density on a set of Lebesgue measure zero.
As a result, as P (Y |Ω) =
∏I
i=1Hi, we estimate our model via maximum likelihood
estimation by maximizing logP (Y |Ω) which is equivalent to:
logP (Y |Ω) = log
I∏
i=1
Hi
=
I∑
i=1
log(Hi)
=
I∑
i=1
log(Hi,1 +Hi,2), (2.20)
where Hi,1 and Hi,2 are defined as above. We state this result as a theorem as it is
the main result of this chapter.
Theorem 2.2.2 (Marginalized Logit Likelihood Assuming Independent Normal
Prior Distributions). The log marginalized likelihood of (2.12) assuming independent
normal heterogeneity distributions, based on a convergent series approximation to
(2.3), is provided by:
21
logP (Y |Ω) =
I∑
i=1
log(Hi,1 +Hi,2), (2.21)
where:
Hi,1 =
∞∑
ki=0
(−1)kie
(yi−1−ki)
2∑P
p=1 x
2
i,pσ
2
p
2 +(yi−1−ki)
∑P
p=1 xi,pµp
·Φ


(yi − 1− ki)
∑P
p=1 x
2
i,pσ
2
p +
∑P
p=1 xi,pµp
√∑P
p=1 x
2
i,pσ2p


(2.22)
Hi,2 =
∞∑
ki=0
(−1)kie
((yi+ki))
2∑P
p=1 x
2
i,pσ
2
p
2 +(yi+ki)
∑P
p=1 xi,pµp
·Φ

−
(yi + ki)
∑P
p=1 x
2
i,pσ
2
p +
∑P
p=1 xi,pµp
√∑P
p=1 x
2
i,pσ2p

 (2.23)
Theorem 2.2.2 provides the marginalized likelihood, and we can estimate our
parameters µp and σ2p for p = 1, ..., P by maximizing the above equation and com-
pute associated p-values to determine the statistical significance of the resulting
estimates. This marginalization reduces the parameter space from one of IP dimen-
sions to 2P dimensions, making model estimation on large data sets considerably
easier.
22
2.3 Simulations
2.3.1 Numerical Simulations
We illustrate the efficacy of our method by performing a series of numerical
simulations. In particular, we conducted a series of simulations for p = 2, 3, and
4 attributes, allowing I = 1000, JNi = 1, and 300 terms in the series expansion
(i.e. 1000 households, 1 category, and 1 occasion). These simulations are a subset
of a larger number of simulations conducted that is available upon request. For
each vector of parameters (µ1,...,µP ,...,σ21,...,σ
2
P ), we performed 25 simulations. For
each such simulate, we simulated I = 1000 values of (xi,1, ..., xi,P ), rescaled them
by dividing by a constant to allow for enough 0/1 variation of yi, numerically ap-
proximated P (Y |Ω) via (2.20), and maximized the resulting marginal likelihood as
a function of Ω.
These simulations were run using MATLAB on an AMD 2.2 GHz Triple Core
Processor with 8 GB of RAM. Our results are summarized in Tables 2.1-2.9, con-
sisting of the true values of (µ1,...,µP ,...,σ21,...,σ
2
P ), the mean and standard deviation
of each of these values, and the corresponding t-statistics. The simulations corre-
sponding to p = 4 attributes is split up into several tables (Tables 2.5-2.9) due to
space constraints. In order to determine whether the values were in numerical cor-
respondence with the true values, we conducted t-tests for each of the parameters.
This resulted in comparing the computed t-statistics to a t-distribution with 24 de-
grees of freedom, as a result of performing 25 simulations for each set of parameter
estimates. After instituting conservative but commonly accepted Bonferroni correc-
tions, we used critical values of 3.791 for p = 2 attributes, 4.051 for p = 3 attributes,
and 4.244 for p = 4 attributes to compare the calculated t-statistics against. All
of our values are well below these critical values. These significance tests therefore
do not suggest any meaningful disparity between the estimated parameter values
23
and the true values. As these runs were based on simulated data from a plethora of
normal distributions, these simulations demonstrate the strong accuracy of our poly-
nomial approximations as well as the efficacy of our new technique. Additionally,
the final few terms of the tails of the truncated series approximations were essen-
tially zero (according to MATLAB output) for a variety of choices of truncation
levels. This fact suggested that, at least for these simulations, the truncated series
expansions based on 300 terms were reasonable approximations to the marginalized
likelihood.
Tab. 2.1: Numerical Simulations for P = 2 attributes
(µ1, µ2) (σ21, σ
2
2) (µ1, µ2) (σ
2
1, σ
2
2) (σµ1 , σµ2) (σσ21 , σσ22)
(-8, -3) (3, 8) (-7.970, -2.972) (2.556, 7.658) (1.581, 1.417) (1.195, 1.387)
(-4, -14) (5, 6) (-3.728, -14.175) (4.772, 6.511) (1.188, 1.236) (1.273, 1.037)
(-7, -5) (5, 3) (-7.300, -5.078) (4.691, 2.710) (1.434, 1.426) (1.582, 1.366)
(5, -4) (4, 5) (4.865, -3.842) (4.265, 5.239) (1.001, 1.210) (1.432, 1.150)
(-5, 4) (4, 5) (-4.801, 4.123) (4.385, 5.072) (1.408, 1.318) (1.252, 1.252)
(5, 4) (4, 5) (5.062, 3.893) (4.429, 5.312) (1.377, 1.279) (1.441, 1.086)
(8, -3) (3, 8) (8.069, -3.232) (2.771, 8.290) (1.742, 1.816) (0.969, 0.636)
(-8, 3) (3, 8) (-7.601, 2.904) (2.942, 8.389) (1.587, 1.414) (0.745, 0.885)
(8, 3) (3, 8) (8.182, 2.970) (2.173, 8.484) (1.329, 1.362) (1.387, 1.873)
(4, -14) (5, 6) (3.779, -14.461) (5.259, 5.584) (1.077, 1.353) (1.242, 1.303)
(-4, 14) (5, 6) (-3.945, 13.877) (5.167, 5.557) (0.945, 1.135) (1.088, 1.012)
(7, -5) (5, 3) (7.061, -4.359) (5.005, 3.416) (1.253, 1.111) (1.686, 1.374)
(-7, 5) (5, 3) (-7.666, 5.115) (4.806, 2.929) (1.036, 1.319) (1.344, 1.542)
(7, 5) (5, 3) (7.327, 4.565) (4.786, 2.678) (1.410, 1.240) (0.858, 1.323)
2.3.2 Comparisons to MCMC Methods
We also assessed the efficacy of our series expansion approach by comparing
the speed of the approach to that of existing MCMC methods [50]. For our MCMC
baseline comparison, we used the bayesm package in R (Rossi and McCullouch
2006) and to ensure a strict “apples-to-apples” comparison of computing times, we
also translated our series expansion approach from MATLAB to R. We ran the
24
Tab. 2.2: t-statistics for P = 2 attributes
(µ1, µ2) (σ21, σ
2
2) t-stat (µ1) t-stat (µ2) t-stat (σ
2
1) t-stat (σ
2
2)
(-8, -3) (3, 8) 0.095 0.098 1.859 1.232
(-4, -14) (5, 6) 1.147 -0.708 0.897 -2.461
(-7, -5) (5, 3) -1.047 -0.272 0.976 1.063
(5, -4) (4, 5) -0.672 0.652 -0.924 -1.040
(-5, 4) (4, 5) 0.706 0.466 -1.538 -0.289
(5, 4) (4, 5) 0.224 -0.418 -1.490 -1.437
(8, -3) (3, 8) 0.197 -0.640 1.180 -2.275
(-8, 3) (3, 8) 1.257 -0.339 0.392 -2.201
(8, 3) (3, 8) -0.683 0.111 2.981 -1.291
(4, -14) (5, 6) -1.025 -1.704 -1.042 1.597
(-4, 14) (5, 6) 0.292 -0.542 -0.765 2.187
(7, -5) (5, 3) 0.242 2.887 -0.013 -1.514
(-7, 5) (5, 3) -3.212 0.435 0.723 0.230
(7, 5) (5, 3) 1.159 -1.756 1.246 1.216
Tab. 2.3: Numerical Simulations for P = 3 attributes
(µ1, µ2, µ3) (σ
2
1 , σ
2
2 , σ
2
3) (µ1, µ2, µ3) (σ
2
1 , σ
2
2 , σ
2
3) (σµ1 , σµ2 , σµ3 ) (σσ1 , σσ2 , σσ3 )
(-5, -6, -7) (3, 4, 3) (-5.010, -6.298, -6.883) (3.268, 3.973, 3.232) (1.591, 1.453, 1.704) (1.037, 1.091, 0.941)
(5, -6, -7) (3, 4, 3) (5.030, -5.690, -7.658) (3.028, 4.098, 2.679) (1.788, 1.897, 1.382) (1.397, 1.128, 1.357)
(-5, 6, -7) (3, 4, 3) (-5.283, 6.043, -6.691) (2.685, 3.947, 3.189) (1.159, 1.369, 1.617) (1.288, 0.748, 0.923)
(-5, -6, 7) (3, 4, 3) (-4.803, -5.962, 7.247) (2.930, 4.096, 2.885) (1.242, 1.541, 1.396) (1.250, 0.882, 1.139)
(5, -6, 7) (3, 4, 3) (4.745, -5.889, 7.093) (3.130, 3.925, 3.263) (1.295, 1.600, 1.501) (1.054, 1.038, 0.766)
(5, 6, 7) (3, 4, 4) (4.964, 6.290, 6.987) (2.972,4.339, 4.123) (1.542, 1.423, 1.593) (0.970, 1.617, 1.397)
(-15, -4, -6) (2, 7, 4) (-15.347, -3.751, -6.367) (2.116, 7.619, 3.472) (2.045, 1.894, 2.162) (1.009, 2.977, 2.961)
(15, -4, -6) (2, 7, 4) (14.352, -3.913, -5.930) (2.005, 7.058, 4.202) (1.708, 1.568, 1.579) (0.825, 2.255, 1.830)
(-9, -8, 4) (3, 3, 5) (-9.203, -7.712, 3.560) (3.024, 3.637, 4.689) (1.567, 1.737, 1.919) (1.011, 1.888, 1.167)
(-15, -4, 6) (2, 7, 4) (-14.975, -3.882, 5.610) (1.888, 6.995, 4.364) (1.890, 1.528, 1.790) (0.981, 2.478, 2.290)
(15, -4, 6) (2, 7, 4) (14.601, -3.780, 6.152) (1.761, 6.503, 4.535) (2.174, 1.919, 1.853) (0.940, 2.714, 2.688)
(15, 4, 6) (2, 7, 4) (14.818, 4.131, 5.782) (2.169, 6.344, 4.371) (1.891, 2.127, 1.729) (0.986, 2.706, 2.787)
(-9, -8, -4) (3, 3, 5) (-9.039, -8.245, -3.967) (2.721, 2.975, 5.219) (1.641, 1.504, 1.533) (1.989, 2.321, 2.394)
(-9, 8, -4) (3, 3, 5) (-9.112, 8.124, -4.107) (3.191, 3.364, 4.949) (1.205, 1.686, 1.434) (1.022, 0.762, 0.598)
(-9, -8, 4) (3, 3, 5) (-9.203, -7.712, 3.560) (3.024, 3.637, 4.689) (1.567, 1.737, 1.919) (1.011, 1.888, 1.167)
(9, -8, 4) (3, 3, 5) (9.699, -7.759, 3.298) (3.104, 2.810, 4.954) (1.591, 1.361, 1.421) (0.557, 0.782, 0.557)
(-9, 8, 4) (3, 3, 5) (-9.176, 7.820, 4.318) (2.949, 3.260, 5.068) (1.440, 1.403, 1.431) (1.157, 1.054, 0.993)
(9, 8, 4) (3, 3, 5) (9.394, 8.074, 3.907) (2.709, 2.310, 4.487) (1.693, 1.531, 1.629) (2.403, 2.344, 2.304)
Tab. 2.4: t-statistics for P = 3 attributes
(µ1, µ2, µ3) (σ
2
1 , σ
2
2 , σ
2
3) t-stat (µ1) t-stat (µ2) t-stat (µ3) t-stat (σ
2
1) t-stat (σ
2
2) t-stat (σ
2
3)
(-5, -6, -7) (3, 4, 3) -0.030 -1.024 0.342 -1.293 0.125 -1.231
(5, -6, -7) (3, 4, 3) 0.083 0.818 -2.382 -0.101 -0.433 1.185
(-5, 6, -7) (3, 4, 3) -1.221 0.156 0.955 1.222 0.352 -1.023
(-5, -6, 7) (3, 4, 3) 0.791 0.123 0.883 0.279 -0.544 0.507
(5, -6, 7) (3, 4, 3) -0.984 0.347 0.310 -0.614 0.364 -1.717
(5, 6, 7) (3, 4, 4) 0.118 -1.016 0.041 0.147 -1.048 -0.442
(-15, -4, -6) (2, 7, 4) -0.848 0.657 -0.849 -0.572 -1.040 0.892
(15, -4, -6) (2, 7, 4) -1.897 0.279 0.223 -0.031 -0.129 -0.552
(-9, -8, 4) (3, 3, 5) 0.648 -0.829 1.146 0.121 1.686 -1.333
(-15, -4, 6) (2, 7, 4) 0.066 0.385 -1.089 0.570 0.010 -0.794
(15, -4, 6) (2, 7, 4) -0.917 0.573 0.409 1.273 0.916 -0.996
(15, 4, 6) (2, 7, 4) -0.481 0.307 -0.630 -0.856 1.213 -0.666
(-9, -8, -4) (3, 3, 5) 0.118 0.814 -0.107 -0.702 -0.054 0.456
(-9, 8, -4) (3, 3, 5) 0.465 -0.369 0.372 0.932 2.392 -0.423
(-9, -8, 4) (3, 3, 5) 0.648 -0.829 1.146 0.121 1.686 -1.333
(9, -8, 4) (3, 3, 5) -2.197 -0.887 2.472 0.936 -1.215 -0.411
(-9, 8, 4) (3, 3, 5) 0.611 0.640 -1.111 -0.222 1.235 0.342
(9, 8, 4) (3, 3, 5) -1.163 -0.242 0.285 -0.605 -1.472 -1.113
25
Tab. 2.5: Numerical Simulations for P = 4 attributes
(µ1, µ2, µ3, µ4) (σ21, σ
2
2, σ
2
3, σ
2
4)
(-5, -14, -3, -2) (3,6,4,6)
(-10,-12,-5,-5) (3,3,3,4)
(-7,-5,-5,-3) (4,6,5,4)
(5,-14,-3,-5) (3,6,4,6)
(-5,14,-3,-5) (3,6,4,6)
(-5,-14,-3,5) (3,6,4,6)
(5,14,-3,5) (3,6,4,6)
(5,-14,3,-5) (3,6,4,6)
(5,14,3,5) (3,6,4,6)
(10,-12,-5,-5) (3,4,3,4)
(-10,12,-5,-5) (3,4,3,4)
(-10,-12,5,-5) (3,4,3,4)
(-10,-12,-5,5) (3,4,3,4)
(10,-12,5,-5) (3,4,3,4)
(-10,12,-5,5) (3,4,3,4)
(10,12,5,5) (3,4,3,4)
(7,-5,-5,-3) (4,6,5,4)
(-7,5,-5,-3) (4,6,5,4)
(-7,-5,5,-3) (4,6,5,4)
(7,-5,5,-3) (4,6,5,4)
(-7,5,-5,3) (4,6,5,4)
(7,5,5,3) (4,6,5,4)
26
Tab. 2.6: Numerical Simulations for P = 4 attributes (continued)
(µ1, µ2, µ3, µ4) (σ21, σ
2
2, σ
2
3, σ
2
4)
(-4.957, -13.727, -3.254, -2.323) (2.368, 6.014, 3.736, 5.074)
(-9.651, -11.951, -5.249, -4.794) (2.753, 3.035, 2.806, 3.984)
(-7.269, -4.924, -5.028, -3.163) (3.523, 5.720, 4.367, 3.839)
(5.088, -14.096, -3.556, -4.840) (2.437, 5.938, 4.067, 6.112)
(-5.618, 14.514, -2.732, -4.960) (2.832, 6.048, 3.760, 5.845)
(-4.707, -14.606, -2.770, 5.084) (2.694, 5.589, 4.116, 6.014)
(-5.120, 14.078, -3.168, 5.304) (3.410, 5.987, 4.348, 5.793)
(4.987, -14.345, 3.140, -4.963) (2.412, 5.604, 3.876, 5.818)
(5.062, 14.542, 2.360, 5.343) (2.597, 5.537, 3.947, 5.476)
(9.624, -11.746, -5.127, -5.109) (2.659, 4.061, 3.088, 3.973)
(-10.161, 12.526, -4.934, -5.120) (3.124, 4.026, 3.069, 4.073)
(-10.167, -12.214, 4.796, -4.899) (2.685, 3.774, 3.066, 3.636)
(-9.792, -11.717, -5.133, 4.834) (3.243, 4.206, 3.034, 3.243)
(10.076, -11.867, 5.134, -4.972) (2.969, 4.024, 2.934, 4.088)
(-9.833, 12.462, -5.085, 4.123) (2.964, 3.948, 3.008, 3.934)
(10.432, 11.902, 5.244, 4.756) (3.054, 4.280, 2.398, 3.640)
(7.045, -5.497, -4.710, -3.204) (4.347, 6.055, 4.975, 3.883)
(-7.097, 5.174, -5.122, -2.976) (3.875, 5.943, 4.816, 3.799)
(-6.715, -5.053, 5.164, -3.247) (4.037, 6.064, 4.898, 4.066)
(7.361, -4.920, 4.834, -2.846) (3.907, 5.966, 5.073, 3.997)
(-7.234, 4.965, -5.234, 3.456) (4.026, 5.923, 5.091, 4.172)
(6.766, 5.419, 5.249, 3.109) (3.757, 5.210, 4.864, 4.174)
27
Tab. 2.7: Numerical Simulations for P = 4 attributes (continued)
(σµ1 , σµ2 , σµ3 , σµ4) (σσ21 , σσ22 , σσ23 , σσ24)
(1.956, 1.762, 1.672, 1.712) (1.231, 1.878, 1.525, 1.599)
(1.938, 1.898, 1.712, 1.827) (2.301, 2.506, 2.298, 2.501)
(1.652, 1.479, 1.743, 1.574) (1.845, 2.094, 2.099, 2.007)
(1.595, 1.482, 1.471, 1.601) (1.593, 0.967, 0.737, 1.153)
(1.277, 1.576, 1.337, 1.517) (1.365, 0.934, 0.749, 0.565)
(1.382, 1.612, 1.570, 1.670) (0.914, 1.146, 0.939, 0.492)
(1.707, 1.542, 1.451, 1.287) (1.502, 0.906, 1.061, 0.875)
(1.440, 1.811, 1.365, 1.349) (1.132, 0.789, 0.786, 0.724)
(1.380, 1.726, 1.839, 1.850) (1.822, 2.091, 2.155, 2.075)
(1.372, 1.680, 1.645, 1.379) (1.313, 0.848, 1.171, 0.761)
(1.588, 1.542, 1.434, 1.666) (0.740, 0.549, 0.887, 0.660)
(1.805, 1.776, 1.723, 1.866) (1.928, 2.108, 2.050, 2.163)
(1.798, 1.769, 1.545, 1.692) (1.936, 1.835, 1.826, 1.678)
(1.656, 1.866, 1.496, 1.623) (0.502, 0.479, 0.235, 0.294)
(1.564, 1.680, 1.755, 1.518) (0.780, 0.575, 0.763, 0.775)
(1.753, 1.815, 1.588, 1.806) (2.352, 2.435, 2.215, 2.432)
(1.138, 1.449, 1.528, 1.654) (1.340, 0.880, 1.124, 1.040)
(1.519, 1.793, 1.945, 1.562) (1.009, 0.880, 0.857, 0.854)
(1.681, 1.850, 1.458, 1.728) (0.469, 0.898, 0.277, 0.689)
(1.294, 1.450, 1.624, 1.433) (0.657, 0.267, 0.215, 0.528)
(1.376, 1.826, 1.736, 1.493) (1.185, 0.630, 0.966, 0.527)
(1.590, 1.671, 1.557, 1.679) (1.903, 2.122, 2.165, 2.243)
28
Tab. 2.8: Numerical Simulations for P = 4 attributes (continued)
t-stat (µ1) t-stat (µ2) t-stat (µ3) t-stat (µ4)
0.111 0.774 -0.758 -0.942
0.899 0.129 -0.727 0.563
-0.813 0.257 -0.079 -0.519
0.276 -0.325 -1.891 0.500
-2.419 1.631 1.003 0.132
1.060 -1.881 0.731 0.250
-0.351 0.254 -0.580 1.180
-0.045 -0.951 0.513 0.138
0.225 1.570 -1.740 0.928
-1.370 0.756 -0.387 -0.396
-0.508 1.705 0.231 -0.360
-0.463 -0.603 -0.591 0.272
0.580 0.799 -0.431 -0.491
0.230 0.356 0.449 0.085
0.534 1.375 -0.241 -2.889
1.233 -0.269 0.768 -0.675
0.199 -1.713 0.949 -0.616
-0.320 0.486 -0.313 0.078
0.848 -0.144 0.562 -0.715
1.394 0.274 -0.512 0.538
-0.849 -0.097 -0.674 1.526
-0.737 1.255 0.801 0.324
29
Tab. 2.9: Numerical Simulations for P = 4 attributes (continued)
t-stat (σ21) t-stat (σ
2
2) t-stat (σ
2
3) t-stat (σ
2
4)
2.568 -0.038 0.866 2.897
0.537 -0.070 0.421 0.032
1.294 0.668 1.508 0.400
1.768 0.324 -0.452 -0.485
0.615 -0.255 1.606 1.372
1.675 1.792 -0.616 -0.147
-1.366 0.070 -1.639 1.183
2.597 2.513 0.790 1.255
1.107 1.106 0.124 1.262
1.300 -0.359 -0.376 0.175
-0.841 -0.236 -0.388 -0.555
0.817 0.537 -0.160 0.841
-0.627 -0.560 -0.094 2.255
0.306 -0.248 1.405 -1.490
0.229 0.455 -0.054 0.425
-0.115 -0.576 1.360 0.741
-1.294 -0.310 0.112 0.563
0.622 0.326 1.072 1.177
-0.392 -0.354 1.841 -0.481
0.708 0.628 -1.689 0.028
-0.108 0.608 -0.471 -1.629
0.638 1.860 0.314 -0.388
30
MCMC sampler for 20,000 iterations, after allowing for 5,000 burn-in iterations. In
particular, we simulated data and estimated our model on this simulated data for
I = 1000 individuals, letting p = 2, 3, or 4 attributes, and JNi = 1. We truncated
our series expansions after 300 terms, which was shown to be quite sufficient in the
previous section. The computing times in minutes for each method is presented in
Table 2.10.
Tab. 2.10: Comparisons of Computing Time for closed-form Series Expansion Approach
versus MCMC Methods (in minutes)
Series Expansions MCMC Methods
p = 2 attributes 2.20 69.20
p = 3 attributes 11.29 69.60
p = 4 attributes 10.98 69.30
As Table 2.10 illustrates, our new technique clearly outperforms existing MCMC
methods. Autocorrelations of draws from the posterior suggested that the number of
iterations performed (20,000 along with 5,000 for burn-in) was necessary in order to
begin having a sense of the entire posterior distribution, although running the sam-
pler for more iterations (and hence for a longer amount of time) would be advisable
in practice. Thus, it is quite clear that our non-iterative series expansion approach
outperforms existing MCMC methods in computing time. These computational
gains can be quite substantial for large data sets.
31
2.4 A Few Generalizations
2.4.1 Incorporating covariances: Positing Heterogeneity from a Multivariate
Normal Distribution
Previously, we had assumed that our parameters βi,1, ..., βi,p were drawn from
independent normal distributions. It is possible, however, to generalize this assump-
tion and allow for correlations amongst the different response coefficients. In some
applications, allowing for correlations is an important assumption to make; for ex-
ample, in modeling consumer choice of products, a consumer’s price sensitivity may
be related to sensitivities to other marketing-related covariates, such as the presence
of a promotion or even the time of year.
Still retaining our initial assumption that all households are independent, we
can allow βi = (βi,1, ..., βi,p) to be drawn from a multivariate normal distribution
with mean µ and variance-covariance matrix Σ:
P (Y |β) =
I∏
i=1
eyiX
T
i βi
1 + eX
T
i βi
,
where (βi,1, ...βi,p) ∼MVN(~µ,Σ). (2.24)
The diagonal elements of Σ represent variances σ21, ...σ
2
p and the off-diagonal
elements ρm,n for m 6= n allow us to model correlations amongst the various coeffi-
cients. The result is a more general integral to evaluate. As we did earlier, we can
define zi =
∑P
p=1 xi,pβi,p. Now that we have covariances in our model, however, it
follows that zi ∼ N(
∑P
p=1 xi,pµp,
∑P
p=1 x
2
i,pσ
2
p +
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn).
Therefore, if we define Pzi as the measure induced by zi on measurable space
(Ti, Gi) having density with respect to Lebesgue measure:
32
f(zi) =
1
√
2pi
∑P
p=1 x
2
i,pσ
2
p +
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn
e
−(zi−
∑P
p=1 xi,pβi,p)
2
2
∑P
p=1 x
2
i,pσ
2
p+
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn ,
then we can again integrate Hi as defined in (2.20):
Hi =
∫ ∞
−∞
[
∞∑
ki=0
(−1)kie(yi−1−ki)ziI(zi > 0)
+
∞∑
ki=0
(−1)kie(yi+ki)ziI(zi < 0) +
eyizi
1 + ezi
I(zi = 0)
]
f(zi)dzi.
We can again decompose Hi into a sum of three integrals, Hi,1, Hi,2, and Hi,3 where
Hi = Hi,1 +Hi,2 +Hi,3. And once again, as a result of Lemma 2.2.1,
Hi,1 =
∫ ∞
0
∞∑
ki=0
(−1)kie(yi−1−ki)zi
e
−(zi−
∑P
p=1 xi,pβi,p)
2
2
∑P
p=1 x
2
i,pσ
2
p+
∑
m6=n,m,n≤P xi,mxi,nρm,nσmσn
√
2pi
∑P
p=1 x
2
i,pσ2p +
∑
m6=n,m,n≤P xi,mxi,nρm,nσmσn
dzi
=
1
√
2pi
∑P
p=1 x
2
i,pσ2p +
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn
∞∑
ki=0
(−1)ki
∫ ∞
0
e(yi−1−ki)zi
·e
−(zi−
∑P
p=1 xi,pβi,p)
2
2
∑P
p=1 x
2
i,pσ
2
p+
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn dzi
Hi,1 =
∞∑
ki=0
(−1)kie
(yi−1−ki)
2(
∑P
p=1 x
2
i,pσ
2
p+
∑
m6=n,m,n≤P xi,mxi,nρm,nσmσn)
2 +(yi−1−ki)
∑P
p=1 xi,pµp
·Φ


(yi − 1− ki)(
∑P
p=1 x
2
i,pσ
2
p +
∑
m6=n,m,n≤P xi,mxi,nρm,nσmσn) +
∑P
p=1 xi,pµp
√∑P
p=1 x
2
i,pσ2p +
∑
m6=n,m,n≤P xi,mxi,nρm,nσmσn

 .
(2.25)
33
Hi,2 =
∫ 0
−∞
∞∑
ki=0
(−1)kie(yi+ki)zi
e
−(zi−
∑P
p=1 xi,pβi,p)
2
2
∑P
p=1 x
2
i,pσ
2
p+
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn
√
2pi
∑P
p=1 x
2
i,pσ2p +
∑
m6=n,m,n≤P xi,mxi,nρm,nσmσn
dzi
=
1
√
2pi
∑P
p=1 x
2
i,pσ2p +
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn
∞∑
ki=0
(−1)ki
∫ 0
−∞
e(yi+ki)zi
·e
−(zi−
∑P
p=1 xi,pβi,p)
2
2
∑P
p=1 x
2
i,pσ
2
p+
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn
Hi,2 =
∞∑
ki=0
(−1)kie
(yi+ki)
2(
∑P
p=1 x
2
i,pσ
2
p+
∑
m6=n,m,n≤P xi,mxi,nρm,nσmσn)
2 +(yi+ki)
∑P
p=1 xi,pµp
·Φ

−
(yi + ki)(
∑P
p=1 x
2
i,pσ
2
p +
∑
m6=n,m,n≤P xi,mxi,nρm,nσmσn) +
∑P
p=1 xi,pµp
√∑P
p=1 x
2
i,pσ2p +
∑
m6=n,m,n≤P xi,mxi,nρm,nσmσn

 .
(2.26)
Hi,3 = 0 as it is once again an integral against a density on a set of Lebesgue
measure zero. As a result, as P (Y |Ω) =
∏I
i=1Hi, we can again estimate our model,
now incorporating correlations amongst the coefficients, via the method of marginal
maximum likelihood by maximizing logP (Y |Ω) which is equivalent to:
logP (Y |Ω) = log
I∏
i=1
Hi
=
I∑
i=1
J∑
j=1
Ni∑
t=1
log(Hi)
=
I∑
i=1
J∑
j=1
Ni∑
t=1
log(Hi,1 +Hi,2). (2.27)
The parameter space is now considerably larger given the need to estimate the off-
diagonal coefficients ρm,n but can still be optimized over using standard optimization
routines.
34
2.4.2 Allowing dependence on other factors
Some researchers may want to allow the mean of βi to depend on certain co-
variates (Zi,1, . . . , Zi,k) that may represent K demographic factors for each particular
individual [141]. Under such a model, we would have:
P (Y |β) =
I∏
i=1
eyiX
T
i βi
1 + eX
T
i βi
,
where (βi,1, ...βi,p) ∼MVN(~∆,Σ), (2.28)
where ~∆ = (
∑K
k=1 Zi,kµ1,k, ...,
∑K
k=1 Zi,kµP,k) and once again, the diagonal elements
of Σ represent variances σ21, ...σ
2
p and the off-diagonal elements ρm,n for m 6= n
allow us to model correlations amongst the various coefficients. The density for
zi =
∑P
p=1 xi,pβi,p with which we integrate our series expansions with respect to is
now a bit different:
zi ∼ N(
P∑
p=1
xi,p
K∑
k=1
Zi,kµp,k,
P∑
p=1
x2i,pσ
2
p +
∑
m 6=n,m,n≤P
xi,mxi,nρm,nσmσn). (2.29)
Therefore, we can now redefine Pzi as the measure induced by zi on measurable
space (Ti, Gi) having density with respect to Lebesgue measure,
f(zi) =
1
√
2pi
∑P
p=1 x
2
i,pσ
2
p +
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn
e
−(zi−
∑P
p=1 xi,p
∑K
k=1 Zi,kµp,k)
2
2
∑P
p=1 x
2
i,pσ
2
p+
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn
then again:
35
Hi =
∫ ∞
−∞
[
∞∑
ki=0
(−1)kie(yi−1−ki)ziI(zi > 0)
+
∞∑
ki=0
(−1)kie(yi+ki)ziI(zi < 0) +
eyizi
1 + ezi
I(zi = 0)
]
f(zi)dzi.
Once again decomposing Hi into a sum of three integrals, Hi,1, Hi,2, and Hi,3 where
Hi = Hi,1 +Hi,2 +Hi,3 and utilizing Lemma 2.2.1,
Hi,1 =
∫ ∞
0
∞∑
ki=0
(−1)kie(yi−1−ki)zi
e
−(zi−
∑P
p=1 xi,p
∑K
k=1 Zi,kµp,k)
2
2
∑P
p=1 x
2
i,pσ
2
p+
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn
√
2pi
∑P
p=1 x
2
i,pσ
2
p +
∑
m6=n,m,n≤P xi,mxi,nρm,nσmσn
dzi
=
1
√
2pi
∑P
p=1 x
2
i,pσ
2
p +
∑
m6=n,m,n≤P xi,mxi,nρm,nσmσn
∞∑
ki=0
(−1)ki
∫ ∞
0
e(yi−1−ki)zi
·e
−(zi−
∑P
p=1 xi,p
∑K
k=1 Zi,kµp,k)
2
2
∑P
p=1 x
2
i,pσ
2
p+
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn dzi
Hi,1 =
∞∑
ki=0
(−1)kie
(yi−1−ki)
2(
∑P
p=1 x
2
i,pσ
2
p+
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn)
2 +(yi−1−ki)
∑P
p=1 xi,p
∑K
k=1 Zi,kµp,k
·Φ


(yi − 1− ki)(
∑P
p=1 x
2
i,pσ
2
p +
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn) +
∑P
p=1 xi,p
∑K
k=1 Zi,kµp,k
√∑P
p=1 x
2
i,pσ
2
p +
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn

 .
(2.30)
36
Hi,2 =
∫ 0
−∞
∞∑
ki=0
(−1)kie(yi+ki)zi
e
−(zi−
∑P
p=1 xi,p
∑K
k=1 Zi,kµp,k)
2
2
∑P
p=1 x
2
i,pσ
2
p+
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn
√
2pi
∑P
p=1 x
2
i,pσ
2
p +
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn
dzi
=
1
√
2pi
∑P
p=1 x
2
i,pσ
2
p +
∑
m6=n,m,n≤P xi,mxi,nρm,nσmσn
∞∑
ki=0
(−1)ki
∫ 0
−∞
e(yi+ki)zi
·e
−(zi−
∑P
p=1 xi,p
∑K
k=1 Zi,kµp,k)
2
2
∑P
p=1 x
2
i,pσ
2
p+
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn
Hi,2 =
∞∑
ki=0
(−1)kie
(yi+ki)
2(
∑P
p=1 x
2
i,pσ
2
p+
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn)
2 +(yi+ki)
∑P
p=1 xi,p
∑K
k=1 Zi,kµp,k
·Φ

−
(yi + ki)(
∑P
p=1 x
2
i,pσ
2
p +
∑
m 6=n,m,n≤P xi,mxi,nρm,nσmσn) +
∑P
p=1 xi,p
∑K
k=1 Zi,kµp,k
√∑P
p=1 x
2
i,pσ
2
p +
∑
m6=n,m,n≤P xi,mxi,nρm,nσmσn

 .
(2.31)
As before, Hi,3 = 0. We utilize these equations to maximize (2.20), which again now
has a slightly larger parameter space as a result of βi,p’s dependence on Zi,k.
2.4.3 Arbitrary Priors
There is no reason to solely restrict ourselves to normal prior distributions. If
we look at the pertinent integration, we can nicely generalize our result provided
we have an analytic expression for the probability density function representing
zi =
∑P
p=1 xi,pβi,p:
Hi =
∫ ∞
−∞
[
∞∑
ki=0
(−1)kie(yi−1−ki)ziI(zi > 0)
+
∞∑
ki=0
(−1)kie(yi+ki)ziI(zi < 0) +
eyizi
1 + ezi
I(zi = 0)
]
f(zi)dzi.
It is easy to notice that this is simply just a linear combination of incomplete mo-
ment generating functions corresponding to f(zi). Thus provided that a closed-form
density for f(zi) exists:
37
Hi =
∞∑
ki=0
(−1)kiIMGF1(yi − 1− ki) +
∞∑
ki=0
(−1)kiIMGF2(yi + ki), (2.32)
where IMGF1 and IMGF2 are the incomplete moment generating functions corre-
sponding to f(zi), integrated from 0 to ∞ and from −∞ to 0 respectively. For an
arbitrary prior, we can use these incomplete moment generating functions to rewrite
our marginalized likelihood and maximize the function accordingly, provided that a
closed-form density corresponding to f(zi) exists.
2.5 Application to Analyzing Potential Causes of Child Poverty
Understanding the potential causes of child poverty is an important issue in pub-
lic policy research. With trillions of dollars being thrown at welfare programs for
decades, it is paramount for policy researchers to analyze potential determinants
of child poverty to evaluate the efficacy of such programs [131]. We do so in this
Chapter.
2.5.1 Data
We used 2009 Current Population Survey (CPS) Data, compiled by the United
States Census Bureau [156]. The CPS data is used for a variety of purposes including
for providing the United States Federal Government’s monthly jobs report. From the
CPS Data, we extracted children under the age of 18 and information of whether
they were below the poverty level (1 if they were below the poverty level and 0
otherwise), education of the head of the household (hereafter referred to as head
parent), marital status of the parents, the head parent’s age, whether the head
38
parent was working full time, and how many people were living in the household.
The resulting data set consisted of slightly more than 60,000 observations.
In existing policy research, these covariates have been considered significant
factors in determining whether a child grows up in poverty [63, 130, 134, 145]. Al-
though logistic regression has been used to analyze poverty data [150], no policy
research, to our knowledge, has looked at this question using models that allow for
individual-level heterogeneity. Allowing for heterogeneity is important as there is
no a priori reason to believe that some individuals would not respond differently
to these factors than others. Furthermore, ignoring heterogeneity can result in the
researcher ignoring potential variability in the model and can therefore lead to mis-
leading statistical inferences.
We therefore used the closed-form expansion derived in the previous sections
to estimate a Bayesian binary logistic regression model with child poverty as a
binary categorical dependent variable and all the other variables mentioned above
as explanatory variables. In order to simplify our model estimation, we assumed no
correlations amongst our coefficients.
Data was rescaled and coefficients were multiplied again by a constant to
ensure for sufficient terms in the series expansion to allow for reasonable approx-
imations. Due to the rapid convergence of the series expansions associated with
this data set indicated by looking the final few terms of the truncated series ap-
proximations for a variety of choices of truncated terms, only 50 terms in our series
expansion were necessary.
2.5.2 Estimation Results
Our estimation results are depicted in Table 2.11. The use of our closed-form
polynomial expansion approach reduced the dimensionality of our problem from
approximately 350,000 parameters to simply just 12 parameters. P-values were
39
determined by performing likelihood ratio tests and utilizing Wilk’s Theorem:
Tab. 2.11: Empirical Bayesian Logistic Regression Estimation Results on Child Poverty
Data
Parameter (µp) p-value (σ2p) p-value
Intercept 0.955 < 0.001 0.000025 < 0.001
Parents Married -2.006 < 0.001 2.902 < 0.001
Head Parent College Educated -1.889 < 0.001 1.869 < 0.001
Head Parent Working Full Time -1.711 < 0.001 2.577 < 0.001
Head Parent Age -0.030 < 0.030 0.000025 < 0.001
Number of Children in the Household 0.345 < 0.001 0.380 < 0.001
Our results shed light on potential factors influencing child poverty. All of our
coefficient estimates are highly significant. In particular, of the covariates looked
at, marital status of the parents is an influential predictor of child poverty, as well
as the educational level of the head parent, and whether the head parent is working
full time. Parental age and the number of children living in the household are also
significant factors influencing whether a child lives in poverty. Our variance esti-
mates also indicate there is a substantial amount of heterogeneity in the population
in how these factors combine to influence child poverty.
The significance of our explanatory variables coincides with common sense -
Two working parents have the potential to bring in more income to a household
than simply just one parent. Additionally, more educated parents have more job
opportunities and therefore have greater potential to comfortably support a family
than less educated parents. Older parents are typically more mature and understand
the challenges associated with having children and often wait until they are ready
to do so. Similarly, more responsible parents will also often wait until they are
financially ready to have additional children.
40
2.5.3 Policy Implications
Our results are in line with research produced by both the Heritage Foundation
and the Brookings Institution [130, 132, 145]. Both think tanks have argued that
marriage is a powerful antidote to child poverty. Recent research, for example, has
argued that marriage in the American society has declined in recent years while
out of wedlock births have steadily increased [130]. In 1964, for example, 93% of
children were born to married parents while in 2007 only 59% of children were born
to married parents. On the other hand, in the mid 1960s, less than 10% of children
were born out of wedlock, while in 2007, this number skyrocketed to 40.6%. As
our results here indicate, children born out of wedlock are overwhelmingly far more
likely to live in poverty than children born to married parents [130,145].
These results have a number of important policy implications. Our results,
along with work from both Heritage and Brookings, suggest that increasing marriage
can significantly reduce child poverty [134]. In order to do so, state and local policy-
makers could consider establishing a campaign of public advertising and education
on teenage abstinence as well as the consequences of child bearing outside of mar-
riage. These campaigns could also work toward communicating the practical issues
faced by single parents, the importance of delaying having children until one is older
and more mature, as well as the importance of waiting until one finds a suitable
partner before doing so [67]. These campaigns could set normative expectations for
younger generations, encouraging young people to become well-educated, to delay
having children until marriage, to work full time to support any children they have,
and to limit their family size to what they can afford [145]. Just as policymakers
have established anti-smoking, anti-drinking, and staying in school campaigns, the
impact of such “pro-marriage campaigns” could be quite significant [130].
Additionally, our results also suggest that welfare programs could benefit from
significant reform. Currently, many means-tested welfare programs such as food
41
stamps, public housing, and Temporary Assistance to Needy Families (TANF) are
structured in a manner that disincentivizes marriage. In particular, many of these
programs have penalties for marriage because welfare benefits decline as a family’s
income rises. Thus, for many low-income mothers, marriage signifies a decline in gov-
ernmental assistance and consequently an overall reduction in the couple’s combined
income. These problems with the welfare system can be ameliorated by reforming
the Earned Income Tax Credit for married couples with children to counteract the
anti-marriage penalties associated with welfare programs [130].
Furthermore, policymakers could also consider strict work and/or study re-
quirements for able-bodied welfare recipients. Such “workfare” requirements not
only have the potential to increase personal income but also encourage only those
truly in need to apply for welfare [129]. Additionally, “workfare” helps enable wel-
fare recipients to acquire valuable training that could be useful in finding subsequent
full time work. In the 1990s, for example, AFDC programs were fundamentally re-
structured into TANF around these ideas, and the result was a rise in employment
as well as a marked decline in child poverty. There are many other American wel-
fare programs that could benefit from similar reforms [135]. The right reforms could
transform many of these programs from the broken safety nets that they currently
are into trampolines that foster growth and success.
2.6 Conclusions and Future Research
With millions of Americans remaining in poverty despite the federal govern-
ment’s endless spending on welfare programs, it is important that policymakers
understand the fundamental causes of poverty. Our study has looked at this issue
and offers a number of informative suggestions. In particular, our results suggest
42
that policies that work toward educating young people about marriage and having a
family could have a significant capacity to reduce child poverty. Additionally, work
requirements for able-bodied welfare recipients could also be particularly helpful.
Methodologically, we now also have an approach for obtaining closed-form
Bayesian inferences for the binary logit model utilizing polynomial expansions, al-
lowing for the use of rich two-sided normal prior distribution. These series expan-
sions can be made arbitrarily close depending on how many terms the researcher
chooses to use in the truncated approximations. Our simulations demonstrate the
efficacy of our technique as well as the fact that our method outperforms existing
MCMC methods. The speed of our approach provides an attractive alternative to
MCMC methods, particularly for large data sets, such as the child poverty data
set used here. Our analysis of this child poverty data set suggests that marriage,
parental age, parental education, and parental work status are significant factors
influencing child poverty. Our findings are in line with extant policy research, and
we suggest a number of policy implications and suggestions based on our results
coupled with these studies.
Although we have focused our application of this model to public policy re-
search, there is no reason that this model cannot be applied to other fields where
logistic regression is used including marketing, economics, biostatistics, criminology,
and professional sports among others. Methodologically, there are also many poten-
tial avenues of future research that this study should encourage. For example, we
made the assumption in this study that JNi = 1. Future research should look into
weakening this restriction. Additionally, a potential avenue of future research is to
explore other methods of polynomial expansions and compare them to the approach
using geometric series expansions here. Furthermore, although we primarily con-
centrated on one particular model - the binary logit model, and one class of priors,
the normal distribution, we hope this study spurs research on closed-form Bayesian
43
inferences for other models as well. In particular, a nice aspect about members
of the exponential family is that each member has a certain conjugate prior. It
could be useful from a computational perspective to use polynomial expansions to
approximate posterior distributions within this family for a choice of priors previ-
ously considered non-conjugate. Additionally, the binary logit model discussed here
belongs to a larger class of generalized linear models (GLMs). A potential avenue
of future research could be to utilize polynomial expansions to allow researchers to
make closed-form Bayesian inferences based on other GLMs. Additionally, deriving
a polynomial expansion approach for the multinomial logistic regression model, a
workhorse model in applied economics research, would also be a worthy endeavor of
future research [58,99].
It is always useful to have a variety of methods to draw inferences from sta-
tistical models, especially for large data sets. We hope the polynomial expansion
approach presented here adds significant value to the applied statistician’s toolbox.
44
Chapter 3: Closed-form Bayesian Inferences for Semiparametric Den-
sity Ratio Modeling with Applications to Tort Reform
3.1 Introduction
3.1.1 Tort Reform
Medical doctors belong to one of the most highly-respected professions in
America. Yet they are at risk of facing unnecessary lawsuits everyday. From sur-
geons to obstetricians to general practitioners, virtually all doctors fear the risk
of lawsuit abuse. These risks have been considered by policymakers on both sides
of the aisle to be an important component in reforming our nation’s health care
system [73].
One aspect of the American health care system that is worthy of attention
is medical malpractice reform. The potential for fraud and abuse of the medical
malpractice system unnecessarily raises health care costs by impacting physician
supply and forcing doctors to engage in defensive medicine [19, 85, 151]. Medical
malpractice reform falls into a larger class of reforms of the civil justice system,
known as tort reform. According to Black’s Law Dictionary, a tort is defined as
“... a legal wrong committed upon the person or property independent of contract
...” [46]. In settling these disputes, compensation may be awarded to the victims.
America has always been a highly litigious country with many frivolous law-
suits and unnecessary abuses of the civil justice system. Such a climate has been
shown to impose hidden costs on consumers as well as on business, including, but
not limited to, the health care, automotive, agricultural, and retail sectors of the
American economy [103]. Tort reform seeks to to reduce tort costs by fundamentally
reforming the civil justice system to prevent abuse, thereby reducing unnecessary
litigation. Recent research has illustrated that tort reform can significantly improve
the business climate in America leading to more jobs, better health care, and a more
prosperous economy [104].
Policymakers have sought to pursue tort reform in a number of ways. One
approach has been to impose monetary caps, which limit the amount of money that
a jury may award a plaintiff. These caps may apply to appeal bonds, non-economic
damages, punitive damages, or monetary damage awards. Other approaches have
been to direct reforms toward other aspects of the civil justice system such as class
action lawsuits, imposing statutes of limitations, and requiring attorney fee limita-
tions among others.
In this chapter, we quantify the impact of medical malpractice reforms by esti-
mating the probabilities of extreme tort losses. Computation of these probabilities,
however, is not an elementary statistical problem as different states exhibit different
degrees of litigiousness. Alaska and Texas have been shown, for example, to be
considerably less litigious than other states such as New York and California [104].
Additionally, different localities within states may also exhibit a certain degree of
heterogeneity. For example, New York City and various towns in upstate New
York may differ how litigious they are. In order to capture this type of state-level
heterogeneity in our model, we improve on existing semiparametric density ratio
estimation methodologies. A series of numerical simulations, along with goodness
of fit computations, illustrates the efficacy of our approach.
46
3.1.2 Bayesian Parametric methods
With the rapid improvements in statistical computing power over the course
of the last three decades, incorporating heterogeneity in parametric models has
become increasingly common in the statistical literature. These days, researchers
can choose from a variety of methodologies such as a parametric Bayesian approach,
a non-parametric Bayesian approach, a finite mixture approach, or a combination
of the finite mixture modeling coupled with Bayesian modeling [74,90,92,140].
Modeling individual-level heterogeneity is often important as there is gener-
ally no a priori reason to assume that all individuals (or observations) in a data
set behave in an identical manner. However, as data sets often contain limited in-
formation about each individual, it is quite difficult, if not impossible, to estimate
models incorporating heterogeneity from a frequentist perspective. The Bayesian
approach allows the statistician to assume individual-level parameters adhere to a
lower dimensional probability distribution and perform statistical inferences based
on the parameters of these lower dimensional distributions either via an empirical
Bayesian approach or a fully Bayesian approach [48,110]. By reducing the problem’s
dimensionality, estimation of these statistical models becomes quite feasible.
In this chapter, we adapt such a Bayesian approach to semiparametric method-
ology used thus far only for frequentist statistical modeling. Our approach enables
these models to accommodate individual-level heterogeneity with high-dimensional
parameter spaces. We discuss this semiparametric methodology in the following
section.
3.1.3 Density Ratio Estimation
Density ratio estimation (DRE) methods were first suggested over thirty years
ago. In their 1997 paper, following Prentice and Pyke (1979) and others, Qin and
Zhang suggested instead of making the typically strict parametric assumptions about
47
distributions of data sets, that a researcher only make assumptions about the ratios
of probability densities (known as tilts) based on subsamples within the data sets.
Qin and Zhang (2005) assumed exponential tilts and recommended estimation of
these models via the method of empirical likelihood [116,122,124,125].
Over the years, DRE has had many applications in statistical research. For
example, Gilbert et al (1999) improved on DRE’s methodologies and applied these
improvements to understanding the efficacy of HIV vaccine trials. Several years
later, Kedem et al (2008) applied DRE to time series forecasting [52, 81]. In par-
ticular, their study used DRE to develop distributional assumptions necessary to
forecast mortality rates for various age groups within the United States. Kedem et
al (2009) and Voulgaraki et al (2012) applied DRE to cancer research [80,160]. Both
studies were able to utilize DRE to determine the significance of certain risk factors
for cancer. In addition to these studies, there have been many other studies that
have utilized the semiparametric benefits of the DRE approach [43,44,80,119,122].
Fokianos and Qin (2008) employed importance sampling in connection with
DRE, which required the generation of artificial data [45]. Prior to that, however, all
research using DRE was based on within-sample data. Specifically, data was always
divided into smaller subsets (such as cohorts) and comparisons would be made
between these subsets. Recent research proposed a innovative adaptation of density
ratio estimation known as “out of sample fusion.” Based on the idea of having one
primary data set as a reference, and a secondary artificial (potentially simulated)
data set, this research illustrated that more accurate inferences can be made about
the primary data set by applying DRE to both samples [78,82,168]. Earlier in this
dissertation, we illustrated that that this methodology can be particularly useful for
making Bayesian inferences when applied to posterior samples.
To date, however, although a few studies have applied Bayesian methods to
empirical likelihood problems, no studies have done so uniting the semiparametric
48
DRE method with Bayesian methods to model individual-level heterogeneity [91,
106,146,165]. We ameliorate this limitation in this chapter. In particular, we adapt
the Bayesian approach mentioned in the previous section to the semiparametric
DRE method to model individual-level heterogeneity. We apply this methodology
to a data set used in a tort reform study conducted by the Pacific Research Institute
to understand overall distributional properties of tort losses throughout the country
[19].
3.2 Problem Formulation
Suppose that we have a data set consisting of i = 1, ..., I samples and define
j = 1, ..., ni observations within each ith sample, such that we have the following P -
dimensional vectors xi,j = (xi,j,1, xi,j,2, ..., xi,j,P ) with
∑I
i=1 ni = N . Let M = I + 1
and define probability density functions gi such that:
xi,j ∼ gi. (3.1)
Additionally, we can also define gI+1 ≡ g as our reference probability density, de-
scribing another sample of size nI+1. Assume the densities gi satisfy the following
relationship regarding their ratios:
gi(xi,j)
g(xi,j)
= w(θi,xi,j), (3.2)
where θi is a vector-valued statistical parameter to be estimated. Without loss of
generality, we assume exponential tilts, defining w(θi,xi,j) ≡ eαi+β
′
ih(xi,j), where we
are currently assuming h : RP → RP . We allow αi ∼ N(µα, 1) and βi ∼ N(µβ,Σβ)
49
where β′i = (βi,1, ..., βi,P ) be our heterogeneity distributions.
1 By allowing our
model’s coefficients to vary for each sample, we enable our model to capture “sample-
level” (or individual-level if each sample represents an individual) heterogeneity.2
We begin by assuming that Σβ is a diagonal matrix and hence that the random
variables βi,p are statistically independent of each other. Additionally, we make the
assumption that ni = 1 ∀ i = 1, . . . , I (to assume a single observation for every
individual in a data set) and nI+1 = nM = N .
3.2.1 Bayesian Density Ratio Estimation
Let G(x) = GI + 1(x) be the reference CDF and define pij = dG(xi,j) =
dGI+1(xi,j). We can utilize the method of constrained empirical likelihood and
estimate gi and θi as follows. We can write the empirical likelihood function, based
on our pooled data xij :
L(θ, GM) =
M∏
i=1
ni∏
j=1
pij
I∏
i=1
ni∏
j=1
eαi+β
′
ih(xi,j), (3.3)
where θ = (α1, . . . , αI , β1,1, . . . , βI,P ).3 As stated above, we make the assumption
that ni = 1 ∀ i = 1, . . . , I and nI+1 = nM = N = I. As a result, the second prod-
uct in the exponential terms does not contribute to multiplication of the distortion
function. We marginalize the empirical likelihood function to generate a “marginal-
ized empirical likelihood” by integrating the above empirical likelihood against the
heterogeneity distributions:
1 For example, one potential choice for h is h(x) = x as in Voulgaraki et al (2012) [160]. This
definition can be generalized, however, and the dimension of β′i would consequently also need to
be altered.
2 We assume the above parameterization for αi (with a constant variance) to ensure statistical
identifiability of the model after marginalization.
3 Chapter 4 contains some theoretical discussion regarding empirical likelihood estimation in-
volving density ratios.
50
ML(µα,µβ,Σβ, GM ) =
∫ +∞
−∞
...
∫ +∞
−∞
L(θ, GM )
I∏
i=1
1
√
2pi
e(
−(αi−µα)
2
2 )dαi
·
P∏
p=1
1
√
2piσ2βp
e
(
−(βi,p−µβp )
2
2σ2βp
)
dβi,p
=
∫ +∞
−∞
...
∫ +∞
−∞
M∏
i=1
ni∏
j=1
pij
I∏
i=1
ni∏
j=1
eαi+β
′
ih(xi,j)
1
√
2pi
e(
−(αi−µα)
2
2 )dαi
·
P∏
p=1
1
√
2piσ2βp
e
(
−(βi,p−µβp )
2
2σ2βp
)
dβi,p
=
M∏
i=1
ni∏
j=1
pij
I∏
i=1
ni∏
j=1
∫ +∞
−∞
. . .
∫ +∞
−∞
eαi+β
′
ih(xi,j)
1
√
2pi
e(
−(αi−µα)
2
2 )dαi
·
P∏
p=1
1
√
2piσ2βp
e
(
−(βi,p−µβp )
2
2σ2βp
)
dβi,p
ML(µα,µβ,Σβ, GM ) =
M∏
i=1
ni∏
j=1
pij
I∏
i=1
ni∏
j=1
eµα+
1
2 eµβ
′h(xi,j)+
1
2h(xi,j)
′Σβh(xi,j) (3.4)
As a result, we have the following theorem:
Theorem 3.2.1 (Marginalized Empirical Likelihood for Density Ratio Model As-
suming Normal Prior Distributions). The marginalized log-likelihood function of
(3.3), assuming normal heterogeneity distributions, is provided by:
LL(µα,µβ,Σβ, GM ) = logML(µα,µβ,Σβ, GM )
=
M∑
i=1
ni∑
j=1
log pij +
I∑
i=1
ni∑
j=1
(µα +
1
2
+ µβ ′h(xi,j) +
1
2
h(xi,j)
′Σβh(xi,j)).
This result is simply due to taking the logarithm of (3.4).
One maximizes the above marginalized likelihood subject to constraints anal-
ogous to those used in Voulgaraki et al 2012 [160]: pij ≥ 0,
∑M
i=1
∑ni
j=1 pij = 1,
and
∑M
i=1
∑ni
j=1 pije
µα+ 12+µβ
′h(xi,j)+
1
2h(xi,j)
′Σβh(xi,j) = 1.4 One can perform the opti-
4 This constraint is easy to see after integration of both sides of the constraint imposed in
51
mization of the empirical likelihood function numerically to estimate µα, µβ, and
Σβ.
3.2.2 Advantages of Marginalization
The marginalization of the empirical likelihood function in equation (3.3)
serves a few important purposes. Firstly, in data sets where ni is small, each ob-
servation will contain very limited information regarding each αi and βi. As a
result, marginalizing the empirical likelihood function by integrating over this high-
dimensional parameter space, enables us to markedly reduce the dimensionality of
the problem, to a considerably less intricate model involving just µα,µβ, and Σβ.
Note that this reduction in dimensionality occurs because the marginalization
essentially transforms our model from one density ratio into another. In particular,
after starting with I different samples, each of size 1, using a sample of size N = I
as a reference, and integrating over the parameter space, the density ratio becomes
another exponential with a slightly different functional form. As a result, this new
density ratio compares two different distributions, each of sample size N .
As the regularity conditions outlined in Fokianos (2004) clearly hold for our
marginalized empirical likelihood, our empirical likelihood estimators are statisti-
cally unbiased and asymptotically normal [43]. A researcher can test the null hy-
pothesis that each coefficient is equal to zero against the alternative that it is non-
zero. As illustrated in Kedem et al (2009), the likelihood ratio of each null model
to the full model asymptotically follows a chi-squared distribution [80].
3.2.3 Derivation of Distributions
We can use the results from our optimization to estimate the distributions.
In particular, we can define γ ≡ λ/2N , where λ is a Lagrange multiplier. We can
Voulgaraki et al (2012):
∑M
i=1
∑ni
j=1 pije
αk+β
′
kh(xi,j) = 1 over the heterogeneity distributions F,
providing us with
∫ ∑M
i=1
∑ni
j=1 pije
αk+β
′
kh(xi,j) dF =
∫
1 dF ∀ k = 1, . . . , I.
52
subsequently replace γ, µα,µβ, and Σβ by their estimators. As a result, following
the derivations in Voulgaraki (2011) [159], estimators of pˆij and Gˆ(x) are provided
by:
pˆij =
1
2N
1
1 + γˆ[eµˆα+
1
2 eµˆ
′
βh(xi,j)+
1
2h(xi,j)
′Σˆβh(xi,j) − 1]
(3.5)
and:
Gˆ(x) =
M∑
i=1
ni∑
j=1
pˆijI(xij ≤ x)
=
1
2N
M∑
i=1
ni∑
j=1
I(xij ≤ x)
1 + γˆ[eµˆα+
1
2 eµˆ
′
βh(xi,j)+
1
2h(xi,j)
′Σˆβh(xi,j) − 1]
. (3.6)
Furthermore, for the “marginalized distribution,” which we will hereafter refer
to as Hˆ, we have:
Hˆ(x) =
M∑
i=1
ni∑
j=1
pˆije
µˆα+ 12+µˆ
′
βh(xi,j)+
1
2h(xi,j)
′Σˆβh(xi,j)I(xij ≤ x)
=
1
2N
M∑
i=1
ni∑
j=1
eµˆα+
1
2+µˆ
′
βh(xi,j)+
1
2h(xi,j)
′Σˆβh(xi,j)I(xij ≤ x)
1 + γˆ[eµˆα+
1
2 eµˆ
′
βh(xi,j)+
1
2h(xi,j)
′Σˆβh(xi,j) − 1]
. (3.7)
For researchers interested in estimating the probability density function of the
sample, kernel density estimators can be constructed by smoothing the increments
of Hˆ [43, 125, 160]. This research in fact illustrated that one can arrive at more
efficient kernel density estimates as a result of combining data. As mentioned earlier
in this dissertation, optimal bandwidth selection for the kernel density estimation
is discussed in detail in Voulgaraki et al (2012) [160].
53
3.3 Numerical Simulations
To demonstrate the efficacy of our approach, we simulated data sets of size
100 from a variety of distributions (gamma, Weibull, and exponential). The gamma
and Weibull distributions were parametrized with shape parameter αi and scale
parameter βi while the exponential distribution was parameterized by αi. These
parameters were drawn from normal distributions, with αi ∼ N(µα, σ2β) and βi ∼
N(µβ, σ2β). This stochastic nature of our distributions’ parameters enabled us to
simulate datasets that, as described earlier, are “heterogeneous” in nature.
Assuming the density ratio to follow (3.2) having linear exponential tilts
w(αi, βi;xi) = eαi+βixi and only one observation per sample (i.e. ni = 1 ∀ i),
we fused this simulated data with a random draw of equal sample size from a spec-
ified distribution that served as a reference distribution. Specifically, we took our
sample of size 100 and fused it with another sample (a reference distribution) of size
100 from a probability distribution (gamma, uniform, or normal) chosen a priori.
When fusing with a gamma distribution, we fit our original sample to a gamma dis-
tribution, estimated the gamma distribution’s parameters via standard maximum
likelihood techniques, and used a sample of size 100 from this gamma distribution as
a reference. When fusing with a uniform distribution, we fused our distribution with
a sample over the range of our original sample’s data and used a sample of size 100
from this uniform distribution as a reference. Finally, when fusing with a normal
distribution, we fused our simulated data with a sample of size 100 from a normal
distribution with sample mean and sample variance equal to that of the original
sample. We then estimated the distribution of both samples from the combined
data using the method of constrained empirical likelihood.
54
Given the limited information per sample in the simulated data set (as ni = 1),
the marginalization of the empirical likelihood discussed in Section 3.2 enables a
significant reduction of the dimensionality of the model’s parameter space (from 200
to 3) and is consequently quite useful for understanding distributional properties
of the data. To quantify our model’s ability to understand these distributional
properties, we used a diagnostic statistic recommended in Voulgaraki et al (2012)
[160]:
R2α,k = 1− exp
[
−
(
xα
m− xα
)k
]
. (3.8)
In the above equation, m is defined as the number of times the estimated
semiparametric cumulative distribution function falls inside the estimated 1 − α
confidence interval obtained from the corresponding empirical cumulative distribu-
tion function, both functions being evaluated at the sample points. Additionally,
k is a prespecified constant, set by the statistician. Since we fused our data with
a reference sample from a known distribution, we estimated (3.8) on our reference
distribution. If our semiparametric density ratio model were inappropriate, then
our estimates would ruin the integrity of this reference distribution, resulting in
inaccurate estimates and consequently poor goodness of fit results. We estimated
(3.8) for our simulations over choices of k = 1 and k = 2.
Following the notation pertaining to (3.2), we refer to our simulated dataset
as x1 = (x1,1, x2,1, . . . , xi,1, . . . , xI,1) (that is, ni = 1 and p = 1 ∀ i = 1, . . . , I and p =
1, . . . , P ) and the sample with which we are fusing as x2 = (x1,2, x2,2, . . . , xi,2, . . . , xI,2).
In the following tables, αˆMLE and βˆMLE denote the maximum likelihood estimators
of x1 when the data are fused with a gamma distribution, x(1,1) and x(I,1) denote
the minimum and maximum values of x1 respectively when the data are fused with
55
a uniform distribution, and x¯1 and s2x1 denote the mean and variance of x1 when
the data are fused with a normal distribution.
Our results are depicted in Tables 3.1-3.3. We notice that our Bayesian ap-
proach clearly fits better than, if not as well as, the existing DRE method as R2
under our standard DRE approach is always less than or equal to R2 under our new
Bayesian DRE approach. Additionally, we also notice that the choice of x2 with
which we fuse our sample with, also affects the goodness of fit results. For example,
at least in these simulations, we notice that fusing our data with a uniform distribu-
tion results in identical results when comparing the DRE approach to the Bayesian
DRE approach. In some cases, such as for our exponentially distributed samples, we
notice an identical goodness of fit (and hence the standard DRE approach is just as
effective); however, for other cases, such as Weibull samples, fusing with a uniform
distribution results in quite poor goodness of fit. These results suggest that when
using out of sample fusion to make statistical inferences about a particular data set,
the researcher should choose from a variety of different distributions from which to
fuse her data with and compare goodness of fit diagnostics to optimally determine
the distribution’s best estimate.
These results demonstrate a number of important points. Firstly, the den-
sity ratios before the marginalization were slightly misspecified to begin with. For
example, the proper ratio in equation (3.2) for a gamma distribution (fused with
another gamma or uniform distribution) should be of the form w(αi, βi,1, βi,2;xi,1) =
eαi+βi,1xi,1+βi,2log(xi,1) whereas we used a density ratio have functional form eαi+βixi
instead. The proper ratios for the other distributions examined are also slightly
different from what was examined here. In reality, however, when presented with a
data set, a researcher generally does not have any detailed specifications regarding
the distribution of the data. Making the simple specification as we have here, with
an elementary linear tilt function and the marginalization of the resulting empirical
56
Tab. 3.1: First Simulation Set - Out of Sample Fusion with Gamma Distribution, Sample
Size I = 100
DRE Bayesian DRE
xi,1 µα σ2α µβ σ
2
β xi,2 R
2
0.95,1 R
2
0.95,2 R
2
0.95,1 R
2
0.95,2
Γ(αi, βi) 10 4 10 4 Γ(αˆMLE , βˆMLE) 0.957 0.976 0.964 0.982
Γ(αi, βi) 30 25 10 25 Γ(αˆMLE , βˆMLE) 0.886 0.917 0.902 0.929
Γ(αi, βi) 10 4 30 4 Γ(αˆMLE , βˆMLE) 0.980 0.994 0.980 0.994
Γ(αi, βi) 100 100 30 100 Γ(αˆMLE , βˆMLE) 0.934 0.975 0.950 0.979
Γ(αi, βi) 30 25 100 25 Γ(αˆMLE , βˆMLE) 0.993 1.000 0.993 1.000
Weibull(αi, βi) 10 4 10 4 Γ(αˆMLE , βˆMLE) 0.829 0.871 0.847 0.882
Weibull(αi, βi) 30 25 10 25 Γ(αˆMLE , βˆMLE) 0.837 0.876 0.856 0.893
Weibull(αi, βi) 10 4 30 4 Γ(αˆMLE , βˆMLE) 0.582 0.552 0.598 0.576
Weibull(αi, βi) 100 100 30 100 Γ(αˆMLE , βˆMLE) 0.637 0.624 0.639 0.626
Weibull(αi, βi) 30 25 100 25 Γ(αˆMLE , βˆMLE) 0.475 0.368 0.502 0.410
Exponential(αi) 10 4 NA NA Γ(αˆMLE , βˆMLE) 0.952 0.974 0.952 0.974
Exponential(αi) 30 25 NA NA Γ(αˆMLE , βˆMLE) 0.944 0.965 0.944 0.965
Exponential(αi) 10 4 NA NA Γ(αˆMLE , βˆMLE) 0.948 0.972 0.948 0.972
Exponential(αi) 100 100 NA NA Γ(αˆMLE , βˆMLE) 0.936 0.971 0.936 0.971
Exponential(αi) 30 25 NA NA Γ(αˆMLE , βˆMLE) 0.965 0.987 0.965 0.987
Tab. 3.2: Second Simulation Set - Out of Sample Fusion with Normal Distribution, Sample
Size I = 100
DRE Bayesian DRE
xi,1 µα σ2α µβ σ
2
β xi,2 R
2
0.95,1 R
2
0.95,2 R
2
0.95,1 R
2
0.95,2
Γ(αi, βi) 10 4 10 4 N(x¯1, s2x1) 0.870 0.905 0.876 0.906
Γ(αi, βi) 30 25 10 25 N(x¯1, s2x1) 0.730 0.738 0.743 0.754
Γ(αi, βi) 10 4 30 4 N(x¯1, s2x1) 0.952 0.984 0.955 0.987
Γ(αi, βi) 100 100 30 100 N(x¯1, s2x1) 0.808 0.827 0.822 0.844
Γ(αi, βi) 30 25 100 25 N(x¯1, s2x1) 0.988 0.999 0.988 0.999
Weibull(αi, βi) 10 4 10 4 N(x¯1, s2x1) 0.881 0.922 0.897 0.936
Weibull(αi, βi) 30 25 10 25 N(x¯1, s2x1) 0.898 0.947 0.922 0.966
Weibull(αi, βi) 10 4 30 4 N(x¯1, s2x1) 0.589 0.550 0.604 0.571
Weibull(αi, βi) 100 100 30 100 N(x¯1, s2x1) 0.664 0.666 0.704 0.723
Weibull(αi, βi) 30 25 100 25 N(x¯1, s2x1) 0.511 0.431 0.514 0.439
Exponential(αi) 10 4 NA NA N(x¯1, s2x1) 0.782 0.824 0.786 0.830
Exponential(αi) 30 25 NA NA N(x¯1, s2x1) 0.853 0.913 0.853 0.913
Exponential(αi) 10 4 NA NA N(x¯1, s2x1) 0.797 0.850 0.801 0.855
Exponential(αi) 100 100 NA NA N(x¯1, s2x1) 0.859 0.910 0.859 0.910
Exponential(αi) 30 25 NA NA N(x¯1, s2x1) 0.832 0.891 0.832 0.891
57
Tab. 3.3: Third Simulation Set - Out of Sample Fusion with Uniform Distribution, Sample
Size I = 100
DRE Bayesian DRE
xi,1 µα σ2α µβ σ
2
β xi,2 R
2
0.95,1 R
2
0.95,2 R
2
0.95,1 R
2
0.95,2
Γ(αi, βi) 10 4 10 4 Unif(x(1), x(N)) 0.960 0.980 0.960 0.980
Γ(αi, βi) 30 25 10 25 Unif(x(1), x(N)) 0.704 0.699 0.704 0.699
Γ(αi, βi) 10 4 30 4 Unif(x(1), x(N)) 1.000 1.000 1.000 1.000
Γ(αi, βi) 100 100 30 100 Unif(x(1), x(N)) 0.791 0.826 0.791 0.826
Γ(αi, βi) 30 25 100 25 Unif(x(1), x(N)) 1.000 1.000 1.000 1.000
Weibull(αi, βi) 10 4 10 4 Unif(x(1), x(N)) 0.308 0.177 0.308 0.177
Weibull(αi, βi) 30 25 10 25 Unif(x(1), x(N)) 0.303 0.170 0.303 0.170
Weibull(αi, βi) 10 4 30 4 Unif(x(1), x(N)) 0.144 0.037 0.144 0.037
Weibull(αi, βi) 100 100 30 100 Unif(x(1), x(N)) 0.143 0.032 0.143 0.032
Weibull(αi, βi) 30 25 100 25 Unif(x(1), x(N)) 0.107 0.020 0.107 0.020
Exponential(αi) 10 4 NA NA Unif(x(1), x(N)) 1.000 1.000 1.000 1.000
Exponential(αi) 30 25 NA NA Unif(x(1), x(N)) 1.000 1.000 1.000 1.000
Exponential(αi) 10 4 NA NA Unif(x(1), x(N)) 1.000 1.000 1.000 1.000
Exponential(αi) 100 100 NA NA Unif(x(1), x(N)) 0.981 0.999 0.981 0.999
Exponential(αi) 30 25 NA NA Unif(x(1), x(N)) 1.000 1.000 1.000 1.000
likelihood function, still enables us to reasonably estimate our distributions’ data,
as our goodness of fit results illustrate. The fact that misspecified tilt functions can
still lead to reasonable distributional estimates has been illustrated in Katzoff et al
(2013) [78].
In the following section, we discuss a few nice extensions of our theoretical
results.
3.4 A Few Generalizations
3.4.1 Arbitrary Prior Distributions and Tilt Functions
Although we assumed normal heterogeneity distributions, there is no a priori
reason to restrict ourselves to this parametric family. In particular, if we generalize
our assumptions and posit an arbitrary prior distribution and tilt function, we notice
that that the marginalized empirical likelihood function simply just involves the mo-
58
ment generating function of the prior measure. Mathematically, we can quite easily
derive this generalization again under the assumption that ni = 1 ∀ i = 1, . . . , I and
nI+1 = nM = N . Following the density ratio in (3.2), and supposing that αi follows
a distribution with density P (αi|θα) with respect to Lebesgue measure and that
each βi,p follows a distribution, statistically independent of αi and of each other,
with density P (βi,p|θβp), we can generalize our marginalized empirical likelihood as
follows:
ML(θα,θβ1 , . . . ,θβP , GM ) =
∫ +∞
−∞
...
∫ +∞
−∞
L(θ, GM )
I∏
i=1
P (αi|θα)dαi
P∏
p=1
P (βi,p|θβp)dβi,p
=
∫ +∞
−∞
. . .
∫ +∞
−∞
M∏
i=1
ni∏
j=1
pij
I∏
i=1
ni∏
j=1
eαi+β
′
ih(xi,j)P (αi|θα)dαi
·
P∏
p=1
P (βi,p|θβp)dβi,p
=
M∏
i=1
ni∏
j=1
pij
I∏
i=1
ni∏
j=1
∫ +∞
−∞
. . .
∫ +∞
−∞
eαi+β
′
ih(xi,j)P (αi|θα)dαi
·
P∏
p=1
P (βi,p|θβp)dβi,p
ML(θα,θβ1 , . . . ,θβP , GM ) =
M∏
i=1
ni∏
j=1
pij
I∏
i=1
ni∏
j=1
MGF (θα; 1)
P∏
p=1
MGF (θβp ;h(xi,j)), (3.9)
where θ = (α1, . . . , αI , β1,1, . . . , βI,P ) as in (3.3). The result is thus in terms of the
moment generating functions of our prior distributions (MGF ) and the tilt function
h(xi,j). The log-likelihood, subject to the appropriate constraints (similar to those
imposed on Theorem 3.2.1), can be optimized to estimate the model. Of course,
prior distributions need to be chosen to ensure statistical identifiability.
59
3.4.2 Assuming homogeneity in the intercepts
To make estimation easier, the researcher may prefer to restrict assumptions
regarding heterogeneity of coefficients corresponding to the explanatory variables
only. We can do so by assuming that the intercepts follow a delta mass at a particular
constant cα such that αi ∼ δ(cα) :
ML(cα,θβ1 , . . . ,θβP , GM ) =
∫ +∞
−∞
...
∫ +∞
−∞
L(θ, GM )
I∏
i=1
δ(cα)dαi
P∏
p=1
P (βi,p|θβp)dβi,p
=
∫ +∞
−∞
...
∫ +∞
−∞
M∏
i=1
ni∏
j=1
pij
I∏
i=1
ni∏
j=1
eαi+β
′
ih(xi,j)δ(cα)dαi
·
P∏
p=1
P (βi,p|θβp)dβi,p
=
M∏
i=1
ni∏
j=1
pij
I∏
i=1
ni∏
j=1
∫ +∞
−∞
. . .
∫ +∞
−∞
eαi+β
′
ih(xi,j)δ(cα)dαi
·
P∏
p=1
P (βi,p|θβp)dβi,p
ML(cα,θβ1 , . . . ,θβP , GM ) =
M∏
i=1
ni∏
j=1
pij
I∏
i=1
ni∏
j=1
ecα
P∏
p=1
MGF (θβp ;h(xi,j)). (3.10)
In the following section, we apply our approach to a fundamental aspect of
the American civil justice system with applications to health care reform.
3.5 An Application: Tort Reform
In this section, we analyze tort loss data across all fifty states. Recent research
has found that tort reform has had a significant impact on reducing insurance premi-
ums as well as on tort losses throughout the country. We utilize the Bayesian DRE
approach in this study to quantify the probability of tort losses exceeding a specified
60
threshold [19] for 2004 and 2006. Computation of these probabilities enables us to
understand on how litigious the country indeed is, thereby shedding light on the
efficacy of recently instituted tort reforms.
3.5.1 Data
Our dataset was the same data used in the Crain et al (2009) study and was
provided to us by two of the paper’s authors [19]. We examined per capita tort
losses defined as the ”payments by defendants (or their insurance companies) for
judgments, settlements, attorney fees, and administrative expenses in tort lawsuits
...”, in thousands of (real 2006) dollars per capita, of each of the fifty U.S. states in
2004 and 2006 [19]. For this analysis, we adhered to examining medical malpractice
tort losses, although analysis of other aspects of the civil justice system (such as
automobile insurance, product liability insurance, and homeowners insurance among
others) are worthwhile endeavors for future research.
Our data is denominated in losses per capita to enable comparable analysis
across the different states. The data set consisted of one observation for each of the
fifty states (for a total sample size of 50 for each year), similar to what we assumed
in the simulation above. As different states and localities throughout the country
have the potential to be more litigious than others, it is important to be able to
capture this heterogeneity in estimating the probability of tort losses exceeding a
particular threshold. This fact, coupled with the fact that our data set contains only
one observation per state, makes our semiparametric Bayesian approach particularly
useful for reducing the dimensionality of the problem.
3.5.2 Estimation
We examined 2004 and 2006 separately, treating each datum within each year
as an independent single-observation sample with its own unique parametrization.
It is worthwhile to compare the distributions of per capita tort losses between 2004
61
and 2006 to understand the efficacy of tort reforms that had been recently insti-
tuted around that time period [19]. In particular, we used the following model
specification:
gi(x)
g(x)
= eαi+βix; i = 1, . . . , 50 (3.11)
and let αi ∼ N(µα, 1) and βi ∼ N(µβ, σ2β) as we did in Section 3.2. By allowing
our model’s coefficients to vary for each state, we enable our model to capture
state-level heterogeneity. We estimated the marginalized distribution by fusing the
sample with data regarding tort losses from 1996. After estimating our marginalized
distribution, we then used the cumulative distribution outlined in (3.7) to compute
the probabilities of extreme tort losses. These probabilities better enable us to
understand the risks associated with litigation across the country.
Additionally, we used a bootstrap approach to develop 95% confidence in-
tervals around these point estimates. Specifically, we resampled our dataset (with
replacement) 1000 times and re-estimated our probabilities for each sample. We used
the resulting set to generate our interval estimates. Our results are are outlined in
Tables 3.4-3.7.
Tab. 3.4: Coefficient Estimates - 2004, Using Bayesian DRE Approach
Coefficient Estimate Lower 95% CI Upper 95% CI
µα -1.3514 -2.0632 -0.5848
µβ 0.0419 -0.0300 0.0765
σ2β 0.0003 0.0000 0.0031
Tab. 3.5: Coefficient Estimates - 2006, Using Bayesian DRE Approach
Coefficient Estimate Lower 95% CI Upper 95% CI
µα 0.4196 -0.2142 1.0331
µβ -0.1102 -0.1896 -0.0368
σ2β 0.0044 0.0066 0.0076
62
Tab. 3.6: Analysis of 2004 Tort Loss Data, Using Bayesian DRE Approach
Probability Estimate Lower 95% Limit Upper 95% Limit
P(Tort Losses > 35000) 0.100 0.010 0.148
P(Tort Losses > 45000) 0.085 0.005 0.104
P(Tort Losses > 55000) 0.019 0.000 0.060
Tab. 3.7: Analysis of 2006 Tort Loss Data, Using Bayesian DRE Approach
Probability Estimate Lower 95% Limit Upper 95% Limit
P(Tort Losses > 35000) 0.068 0.010 0.144
P(Tort Losses > 45000) 0.045 0.005 0.102
P(Tort Losses > 55000) 0.019 0.000 0.060
We also estimated the goodness of fit diagnostic statistics outlined in Table
3.8 for k = 1 and k = 2, comparing it to the existing DRE method. These results
are outlined in Table 3.8. Additionally, Figures 3.1-3.4 depict plots comparing the
estimated distributions of per capita tort losses Hˆ from (3.7) versus the correspond-
ing empirical CDFs for 2004 and 2006. The plots suggest that the comparative fit
between the two models is quite comparable in 2004 (as there is near perfect agree-
ment between the estimated and empirical CDFs) but substantial improvement as
a result of the Bayesian approach compared to the standard approach in 2006.
Tab. 3.8: Goodness of Fit Diagnostics
DRE Bayesian DRE
R20.95,1 R
2
0.95,2 R
2
0.95,1 R
2
0.95,2
2004 0.915 0.998 0.911 0.997
2006 0.108 0.129 0.607 0.582
Our results illustrate that the Bayesian DRE approach substantially improved
model fit in 2006 suggesting that there was unobserved heterogeneity across the
country during that year that the standard DRE approach was unable to properly
model. Although there are not really any noticeable differences between the two
approaches in 2004 (in fact the standard DRE model performs slightly better), this
63
Fig. 3.1: Plot of Hˆ vs. H˜ - DRE, 2004 Per Capita Tort Loss Data
Fig. 3.2: Plot of Hˆ vs. H˜ - Bayesian DRE, 2004 Per Capita Tort Loss Data
64
Fig. 3.3: Plot of Hˆ vs. H˜ - DRE, 2006 Per Capita Tort Loss Data
Fig. 3.4: Plot of Hˆ vs. H˜ - Bayesian DRE, 2006 Per Capita Tort Loss Data
65
similarity can be explained by the fact that in 2004, the 50 states may have been
considerably more homogeneous in nature regarding litigation. In terms of per
capita tort losses, the probabilities of extreme losses clearly declined between 2004
and 2006.
In terms of actual tort losses, we notice a reduction in the probabilities of
extreme losses in 2006 compared to 2004. These results, in conjunction with the
results from Crain et al (2009), illustrate the efficacy of the number of state-based
medical malpractice reforms, many of which had been instituted around this time
period [19]. Such reforms have included attorney fee limitations, requirements for
pre-trial screening, standards regarding expert witnesses, imposition of strict statute
of limitations for filing lawsuits, and economic damage caps. These results may also
be explained by the fact that existing tort laws, not necessarily recently instituted,
may have become more stringently enforced in 2006 compared to 2004. As a side
note, although the coefficient estimates of σ2β appear to be low, they should not be
taken to imply that there is no heterogeneity in the model. As the units of our
data are in thousands of dollars per capita, seemingly small per capita differences
between states imply substantial variation in litigiousness amongst the states.
3.5.3 Policy Implications
A 2004 study conducted by the non-partisan Congressional Budget Office
found that state-based tort reform reduced the number of lawsuits filed, decreased
the number of damage awards, and lowered insurance claims [17]. Our results above
also illustrate that the risks associated with the civil justice system notably de-
clined between 2004 and 2006. These reductions were primarily due to state-based
tort reforms that were instituted throughout the country, including economic caps,
appeal-bond caps, and standards regarding expert witnesses [19].
Although these results illustrate the efficacy of certain malpractice reforms
66
instituted around this time period, considerably more can be done to reform the
civil justice system including passing tort reforms in states where important laws
are lacking and more stringently enforcing existing laws [19, 103]. Scholars at the
Heritage Foundation, the Cato Institute, the American Enterprise Institute, and
the Brookings Institution have discussed the benefits of malpractice reform, noting
that such reforms have the potential to improve the environment for physicians to
practice, benefiting patients in the process [18,93,139,142].
These benefits are quite well-known. For example, a 2005 study published
in the Journal of the American Medical Association found that malpractice reform
increased physician supply, particularly with specialties associated with high mal-
practice costs [86]. The authors found that without appropriate reforms, the medi-
cal malpractice climate had been encouraging early retirements in and discouraging
entry to particularly risky specialties such as obstetrics, anesthesiology, radiology,
and surgery. Additionally, such an environment results in the practice of defensive
medicine, causing physicians to order unneeded tests due to the fear of lawsuits,
unnecessarily increasing health care costs [85, 151].
3.6 Conclusions and Future Research
Our study illustrates the efficacy of medical malpractice reform in reducing the
probabilities of extreme tort losses. Medical malpractice reform is of course by no
means a “silver bullet” for fixing our nation’s broken health care system. Such
reforms, however, are a very helpful component of the supply side of health care.
On the demand side, policymakers should seek to instill free market competition in
the industry, which will notably reduce costs and improve quality of care [13,20,21]
Methodologically, our study provides a notable contribution to both Bayesian
67
estimation methodologies as well as to semiparametric modeling. In particular, we
have applied empirical Bayesian methods to semiparametric density ratio modeling,
allowing statisticians to incorporate individual-level heterogeneity in such models.
An interesting aspect of this approach is that our marginalization yields a closed-
form expression allowing us to make direct statistical inferences about the population
without having to resort to the numerical approaches typically concomitant with
Bayesian methods. As discussed in Chapter 2, these numerical approaches such as
MCMC methods, can be quite computationally intensive, particularly for large data
sets with high-dimensional parameterizations [50].
Additionally, although we focused our application on medical malpractice re-
form, there is no reason that this approach cannot be used in other settings where
modeling individual-level heterogeneity is important without being forced to adhere
to strict parametric assumptions. In particular, the approach outlined here can be
useful in many application areas including medical research [160], economics [56],
understanding athletic performance [24, 107], and other areas within public policy
research [19, 136] among others. Regardless of the application, we believe that the
Bayesian approach outlined here can be yet another useful tool for statisticians to
use.
68
Chapter 4: Bayesian Inferences of Welfare Reform with Improved
Credible Interval Estimation via Semiparametric Out of
Sample Fusion
4.1 Introduction
4.1.1 “Ending Welfare as We Know it”
Welfare reform is a hot topic of debate amongst policymakers in America [61,
135,152]. The welfare state can be defined as a government system aimed at helping
the disadvantaged by providing benefits such health care, pensions, and financial
programs aimed to help those in need [14]. The first Chancellor of Germany, Otto
von Bismarck, constructed the beginnings of the modern European welfare state
in the late 1800s, creating programs such as old age pensions, accident insurance,
and medical care. In the early 20th century, the welfare state came to Britain with
the introduction of pension programs, free school meals, and unemployment and
health benefits. Shortly afterwards, the welfare state began to grow throughout
other European countries [14].
Although the United States does not have an overarching welfare state that
many European countries do, the country does offer a variety welfare programs [14].
Initiated in the United States in the early 1900s, federal welfare programs were
designed with the intention of operating as safety net programs to aid the disad-
vantaged. President Theodore Roosevelt, for example, proposed a number of such
ideas as part of his ‘New Nationalism’ platform. During that time period, Roosevelt
believed it was incumbent upon the federal government to institute programs such
as a National Heath Service; social insurance for the unemployed, disabled, and
elderly; and to provide workers’ compensation to those with certain work-related
injuries [84].
As a component of his New Deal, President Franklin Delano Roosevelt in-
stituted a number of federal cash assistance programs to assist the poor. These
programs included Social Security, Aid to Dependent Children (ADC), and unem-
ployment compensation programs among others. Several decades later, President
Lyndon Johnson instituted the War on Poverty to further expand America’s welfare
state with the goal of eradicating poverty. In particular, he expanded President
Roosevelt’s ADC programs (which by this time were known as Aid to Families with
Dependent Children or AFDC programs), began food stamp programs, public hous-
ing, Medicare, and Medicaid, all of which were intended to be “safety nets” for those
in poverty [14].
In Losing Ground, Charles Murray of the American Enterprise Institute rig-
orously evaluated the effects of these types of welfare programs [111]. He concluded
that without doubt many of these programs had the potential to foster dependency
and cripple those they intend to help. Murray’s work shattered much of the political
capital advocates had for open-endedly expanding America’s welfare state.
In 1992, Presidential Candidate Bill Clinton campaigned to “end welfare as
we know it,” [16]. Two years later, Congressional Republicans made welfare reform
a fundamental component of their Contract with America [53]. The goal of these
reforms were to transform welfare away from a system that encouraged endless
dependence on taxpayer dollars and instead transform it into a system providing
70
temporary assistance to get people back to work and become active contributing
members of society.
In 1996, Congressional Republicans worked with President Clinton to reform
AFDC Programs [53]. Intended to operate as a safety net for those in poverty,
AFDC programs were simply a government handout to qualifying single mothers.
The government mailed checks to recipients who had virtually no responsibilities in
return. In 1996, President Clinton signed into law the Personal Responsibility and
Work Authorization Act (PROWA), restructuring AFDC programs into a more re-
strictive program known as Temporary Assistance to Needy Families (TANF) [118].
Under TANF, state governments were required to impose federal work standards
on welfare recipients. In particular, TANF required recipients to work or study as
a condition for receiving welfare. In addition, TANF also limited receipt of welfare
benefits to a five year time frame. These requirements, although not particularly
rigorous, were intended to transform the welfare program from a system that fos-
tered dependency into a springboard that gave the disadvantaged temporary help
to become contributing members of society [137].
It is important for policymakers to be able to rigorously examine such funda-
mental changes to government programs. Many reforms, including PROWA, have
been constructed to provide states a certain degree of flexibility in implementing the
law. As a result, different states have the potential to manifest different responses
to the law. Although a number of policy studies have made some quantitative
evaluations of PROWA, no studies, to our knowledge, have done so capturing this
state-level heterogeneity in the associated statistical models [61,132,152].
We perform such an analysis in this study. In particular, we statistically
evaluate the success of PROWA in helping to get people back to work. We do
so by estimating hierarchical Bayesian linear models to rigorously compare AFDC
programs (before welfare reform) to TANF programs (after welfare reform). We find
71
that TANF was considerably more successful than AFDC in getting people back to
work, illustrating the success of the PROWA.
In the process of our analysis, we improve on existing Bayesian statistical es-
timation techniques. In particular, standard Bayesian estimation techniques suffer
from a number of significant limitations. Specifically, standard Markov Chain Monte
Carlo (MCMC) methods used to estimate Bayesian statistical models can require
a significant amount of time to adequately sample a posterior density, especially
for large data sets involving high-dimensional parameter spaces [50]. Furthermore,
despite diagnostic checks, it is difficult to truly know whether an MCMC sampler
has adequately sampled the parameter space. As a result, posterior inferences based
on MCMC sampling may or may not necessarily reflect reality. We provide a novel
adaptation of a semiparametric estimation technique, used thus far only for frequen-
tist statistical models, to more accurately summarize the posterior by providing more
precise estimates of the density’s credible intervals [160].
The contribution of this study is therefore twofold. Firstly, we statistically
examine the efficacy of the welfare reform of the 1990s. Secondly, we improve
on standard Bayesian estimation techniques for hierarchical linear models. As we
discuss, our improvements to Bayesian computation are useful for a variety of models
beyond just the hierarchical linear model structure studied here.
4.2 Statistically Modeling the Impact of Welfare Reform
4.2.1 A Hierarchical Bayesian Model Allowing for State-Level Heterogeneity
We are interested in understanding the efficacy of the Personal Responsibility
and Work Opportunity Reconciliation Act of 1996 [118]. This law imposed a five
72
year time limit on welfare benefits and imposed some, albeit not particularly oner-
ous requirements, on individuals to either work or study as a condition of receiving
welfare. To understand the efficacy of this reform, we looked at the effect of the
total number of welfare caseloads in each state on the number of employment re-
lated discontinuances of receiving welfare. Mathematically, this relationship can be
modeled over i ∈ {1, . . . , 53} territories (50 states as well as Guam, Puerto Rico,
and the District of Columbia) (units) over t ∈ {1, . . . , T} years.
Consider a data set consisting of i ∈ {1, . . . , I} observations (in our case
state/territories) over t ∈ {1, . . . , T} occasions. We define our dependent vector
valued variable to be yi = (yi,t)Tt=1, which represents the welfare caseload discontin-
uances due to employment for state/territory i at time t. We define our explanatory
vector valued variable, xi = (xi,t′)T
′
t′=1, as a vector of dimension equal to yi represent-
ing the total number of welfare caseloads at time t′ (for t′ < t) as we wish to relate
yi,t to past covariates. Thus, xi represents a lagged number of welfare caseloads.
As the PROWA imposed a new five year time limit for TANF recipients, which had
previously not been present for AFDC programs, we conducted our analysis for lags
of 1,2, 3, 4, and 5 years.
73
yi ∼ MVN(βi,0 + βi,1xi,1,Σ) (4.1)
βi,0 ∼ N(µ0, σ
2
0) (4.2)
βi,1 ∼ N(µ1, σ
2
1) (4.3)
Σ−1 ∼Wishart(Φ0, ν0) (4.4)
µ0 ∼ N(0, 10) (4.5)
µ1 ∼ N(0, 10) (4.6)
σ−20 ∼ Γ(10, 10) (4.7)
σ−21 ∼ Γ(10, 10) (4.8)
Σ is a T -dimensional variance-covariance matrix that enables us to capture
cross-year correlations (for example caseloads reductions in one particular state
in one year may have a strong correlation with caseload reductions in that same
state the following year). Another way of viewing this model would be to esti-
mate a hierarchical Bayesian linear model defined as yi,t = βi,0 + βi,1xi,t + i,t for
i = 1, . . . , I, t = 1, . . . , t − 1 with the prior structure discussed above and with
non-zero covariances between i,t′ and i,t ∀t, t′ ≤ T .
To ensure statistical identifiability of our model, we requred β1,0 ≡ 0.1 In
the next section, we discuss estimation of this model, including improvements to
estimating the posterior density’s credible intervals.
4.2.2 The Limitations of Bayesian Computational Methods
Typically, these types of hierarchical Bayesian models lack an analytic form for
their posterior functionals and are therefore estimated numerically using standard
1 This coefficient pertains to the intercept regarding Alabama as our data was organized alpha-
betically.
74
Markov Chain Monte Carlo (MCMC) methods [50]. Researchers use the resulting
MCMC samples to generate a variety of statistical estimators such as posterior
means, posterior standard deviations, and endpoints depicting credible intervals.
There are a number of limitations associated with MCMC sampling, however.
In particular, MCMC sampling can be computationally intensive, particularly for
models involving high-dimensional parameter spaces. Recent research has developed
alternative approaches for Bayesian estimation that attempt to address this issue.
In particular, there are approaches using variational approximations to posterior
densities, polynomial expansion approaches, and alternative sampling approaches
among others [9, 10,39,57,108].
Another limitation of MCMC methods is that although there are some eval-
uatory tools (e.g. Gelman and Rubin 1992) to assess convergence of the chain,
these measures are simply just diagnostics, and it is difficult to determine with
certainty if the MCMC chain has sufficiently navigated the posterior distribution’s
parameter space [51]. Although MCMC samplers theoretically do converge under
reasonable conditions, this convergence is only guaranteed to occur asymptotically
over an infinitely large number of draws [154]. In reality, however, the Bayesian
MCMC samplers need to eventually be truncated. As a result, researchers are of-
ten rightfully concerned with whether their posterior sample based on truncation
of the MCMC sampler truly represents the posterior density. A classic pathologi-
cal example demonstrating this phenomenon involves MCMC sampling of the so-
called “witch’s hat,” where the MCMC sampler can get stuck in a particular area
of the posterior distribution [141]. Premature truncation of the MCMC sampler
can consequently result in misleading statistical inferences regarding the posterior
distribution.
We can mitigate this problem considerably by applying a semiparametric
methodology, known as density ratio estimation, to posterior samples [123–125,160].
75
This estimation technique can enable a researcher to more accurately estimate cred-
ible intervals. Although this semiparametric approach has been used in a variety
of settings, no researchers, to our knowledge, have applied this method to Bayesian
estimation.
4.2.3 Statistical Inferences Based on Density Ratio Estimation
Density ratio estimation (hereafter referred to as DRE) is a semiparametric
statistical estimation technique. This technique has been applied to many settings
including AIDS vaccine trials, the analysis of variance, mortality rate prediction,
cluster detection, and cancer research among others [43,44,52,80,81,83]. In the one
dimensional case, there are i = I + 1 random samples xi with sample size ni such
that
∑I+1
i=1 ni = n:
xi = (xi,1, . . . , xi,ni),
with probability density functions gi, such that:
xi,j ∼ gi, i = 1, . . . , I, I + 1, j = 1, . . . , ni. (4.9)
In utilizing this method, the statistician a priori assumes that gI+1 ≡ g defines
a reference probability density with a particular ratio between gi and g ∀ i = 1, . . . , I.
In many cases, this ratio is an exponential involving a vector-valued function h(x),
known as a tilt function:
gi(x)
g(x)
= eαi+β
′
ih(x). (4.10)
If both gi(x) and g(x) are densities belonging to the exponential family having
functional form:
76
g(x,θ) = d(θ)S(x)exp
[
J∑
j=1
cj(θ)Tj(x)
]
= exp
[ J∑
j=1
cj(θ)Tj(x) + log
[
d(θ)S(x)
]
]
, (4.11)
where θ is a parameter to be estimated, then the densities of any two such members
will clearly have a ratio satisfying (4.15), as will the ratio of a single truncated
such member coupled with the density of a uniformly distributed random variable.
Some examples of special cases resulting in commonly used tilt functions are h(x) =
(x, x2)′ (appropriate for normally distributed data):
gi(x)
g(x)
= eαi+βix+γix
2
, (4.12)
h(x) = (x, log(x))′ (appropriate for data coming from a gamma distribution), and
gi(x)
g(x)
= eαi+βix+γilog(x), (4.13)
h(x) = (xτ , log(x))′ (appropriate for potentially more heavily-skewed data coming
from a generalized gamma distribution):
gi(x)
g(x)
= eαi+βix
τ+γilog(x). (4.14)
Parametric assumptions regarding densities, however, are sufficient but not
necessary conditions for density ratios, such as those above, to be able to properly
model real-world phenomena. Recent research, for example, has shown that the
distributions described in (4.13) and (4.14) are useful in modeling situations where
the data does not necessarily adhere to the strict parameterizations described by
known parametric distributions [78,82,168].
77
4.2.4 Empirical Likelihood Estimation in Semiparametric Density Ratio Models
Suppose we assume the density ratio outlined in (4.15) with tilt function h(x),
gi(x)
g(x)
= eαi+β
′
ih(x). (4.15)
If we let G(x) be the reference CDF and define probability masses pij =
dG(xi,j) = dGI+1(xi,j), then we can utilize the method of constrained empirical
likelihood to estimate gi and the associated parameters as follows. We can write
our empirical likelihood function, parametrized by θ = (α1, . . . , αI ,β1, . . . ,βI) (a
vector of dimension Id), based on our pooled data:
L(θ, G) =
I+1∏
i=1
ni∏
j=1
pij
I∏
i=1
ni∏
j=1
eαi+β
′
ih(xi,j). (4.16)
The resulting log-likelihood is given by:
l = log L =
I+1∑
i=1
ni∑
j=1
log pij +
I∑
i=1
ni∑
j=1
(
αi + β′ih(xi,j)
)
, (4.17)
subject to the following constraints:
pij > 0,
I+1∑
i=1
ni∑
j=1
pij = 1,
I∑
i=1
ni∑
j=1
pije
αk+β′kh(xi,j) = 1, for k = 1, . . . , I. (4.18)
The following conditions, as discussed in Fokianos (2004) and Qin and Law-
less (1995) as cited by Voulgaraki (2011) ensure the existence of an empirical
likelihood estimator [43, 123, 159]. Specifically, if we let f(x, θˆ) = (eαˆ1+βˆ1h(x) −
1, . . . , eαˆI+βˆIh(x) − 1)′, then given the following conditions, there exists, with prob-
ability approaching one, an extremum in a ball around the true parameter vector
θ0:
78
Lemma 4.2.1. Under the assumptions that:
(a) E(f(x,θ0)f ′(x,θ0)) is positive definite,
(b) ∂f∂θ is continuous in a ball around the true value of θ0
(c) ||∂f∂θ || and ||f(x,θ)||
3 are bounded by an integral function with respect to G
in this ball,
(d) E
(∂f
∂θ
)
is of rank Id,
where E is the expectation with respect to the probability measure corresponding to
G, then the coefficients αi and βi, as well as the discrete estimators of Gi(x) ∀ i =
1, . . . , I+1 can be estimated by optimizing (4.17) subject to the constraints outlined
in (4.18) via a two step estimation procedure utilizing the method of Lagrange
multipliers. Specifically, if we can define µk ≡ λk/n, where λk is the pertinent
Lagrange multiplier we find the following estimators, pˆij and Gˆi(x) for i = 1, . . . , I+
1, as follows:
pˆij =
1
n
1
1 +
∑I
k=1 µˆk[e
αˆi+βˆ′ih(xij) − 1]
(4.19)
and:
GˆI+1(x) =
I+1∑
i=1
ni∑
j=1
pˆijI(xij ≤ x)
=
1
n
I+1∑
i=1
ni∑
j=1
I(xij ≤ x)
1 +
∑I
k=1 µˆk[e
αˆi+βˆ′ih(x) − 1]
. (4.20)
We can generalize this result to distributions Gˆi for i = 1, . . . , I to generate
the following estimators:
79
Gˆk(x) =
I+1∑
i=1
ni∑
j=1
pˆije
αˆk+βˆ′kh(x)I(xij ≤ x)
=
1
n
I+1∑
i=1
ni∑
j=1
eαˆk+βˆ
′
kh(x)I(xij ≤ x)
1 +
∑I
k=1 µˆk[e
αˆi+βˆ′ih(x) − 1]
, for k = 1, . . . , I. (4.21)
Furthermore, estimators of the probability densities gi(x) can be obtained via
kernel density estimation applied to the jumps of Gˆi ∀ i = 1, . . . , I + 1 . The inter-
ested reader should refer to Voulgaraki et al (2012), which provides a methodology
for a complete discussion of the approach including techniques for determining the
optimal bandwidth needed for accurate kernel density estimation [160].
Additionally, as long as the following properties hold, our estimators are statis-
tically unbiased and asymptotically normal as illustrated in the following theorem.
In particular, by defining the vector µ ≡ (µ1, . . . , µI) as the Lagrange multipliers
for the optimization of the empirical likelihood function subject to the constraints
outlined in (4.18) (just as has been done above) and by letting µ0 denote the true
value of µ, then, we can state the following theorem:
Theorem 4.2.2. Suppose the following conditions hold:
(a) ∂
2f
∂θ∂θ′ is continuous in a ball around the true value of θ0
(b) There exists an integrable function with respect to G bounding || ∂
2f
∂θ∂θ′ ||
(c) The four conditions outlined in Lemma 4.2.1 hold.
Then:
√
n
(
θˆ − θ0
µˆ− µ0
)
⇒ P, (4.22)
where ⇒ denotes weak convergence to a probability measure P, induced by a mul-
tivariate normal random vector with mean 0 and a particular variance-covariance
80
matrix Σ.
The details of the structure of Σ are complicated and derived in detail in
Fokianos (2004). Lu (2007) contains a comprehensive proof of Theorem 4.2.2 [43,96].
One of the most appealing aspects about this semiparametric approach is that
it can more accurately determine probability distributions from data compared to
its more restrictive parametric counterparts. This advantage is due to the fact
that the combined sample of larger size is used rather than any particular sample
consisting of a smaller number of elements. In Section 4.2.2, we discussed many of
the limitations of present day standard Bayesian computational methods. In the
following section, we illustrate how this semiparametric density ratio technique can
be used to ameliorate these limitations.
4.2.5 Numerical Simulations
In this section, we present a series of numerical simulations to illustrate the
efficacy of using DRE to generate accurate estimates of percentiles from a variety
of different samples. In particular, we applied the DRE method, assuming the ratio
described in (4.12) through (4.14) to samples from known parametric distributions
(normal, student-t, gamma, and Weibull) to determine the samples’ 2.5th and 97.5th
percentiles. Since the supports of these distributions are infinite, fusing a sample
from any of these distributions with a uniformly distributed sample signifies that
the density ratio model holds approximately.
Specifically, we applied the DRE approach outlined in Section 4.2.3 with I = 1
(i.e. 2 samples), taking a sample from one of our known distributions, given a
particular choice of parameters, and fused this sample with a random sample drawn
from a uniform distribution with support including this sample’s entire range. This
uniformly distributed sample served as our reference distribution. This methodology,
known as “out of sample fusion,” was first introduced by Zhou (2013) as a tool for
81
estimating small probabilities [168].
We used the DRE approach to estimate the 2.5th and 97.5th percentiles
(DREk,2.5 and DREk,97.5) of our original distribution over the course of k = 1, . . . , K
simulations. We compared these estimates to the true (2.5th and 97.5th) percentiles
of our original population, Qk,2.5 and Qk,97.5, by computing the squared difference
between the two.
We performedK = 100 such simulations for each choice of parameters, summed
these squared differences for each percentile, and computed averages across all sim-
ulations. We hereafter refer to these mean squared differences as MSEDRE,2.5 and
MSEDRE,97.5:
MSEDRE,2.5 =
∑K
k=1 (DREk,2.5 −Qk,2.5)
2
K
(4.23)
and:
MSEDRE,97.5 =
∑K
k=1 (DREk,97.5 −Qk,97.5)
2
K
. (4.24)
The use of known parametric distributions enables us to understand the effi-
cacy of using DRE for estimating low and high level quantiles of distributions. As
a result, these simulations shed light on the usefulness of using the DRE approach
for credible interval estimation of Bayesian regression coefficients.
Zhou (2013) also developed an approach for quantifying the efficacy of using
out of sample fusion for estimating probabilistic thresholds [168]. Zhou’s work found
that out of sample fusion results in shorter confidence intervals compared to using
empirical distributions solely based on within sample data. These confidence inter-
vals, however, are inherently frequentist in nature, based on the assumption that
each sample is the single realization of a long-run frequency of an asymptotically
large number of samples. This approach, although not completely useless, is not
82
philosophically compatible with the Bayesian philosophy of statistical estimation.
As a result, although such an approach sheds light on the efficacy of out of sample
fusion, the approach presented here provides further verification as well as a more
legitimate basis for applying the DRE method to Bayesian estimation.
Our results are outlined in the Tables 4.1-4.10, which we conducted for sam-
ples of size 600, 1000, 5000, 10000, 15000, 20000, and 25000. DRE2.5 and DRE97.5
represent the estimates of the 2.5th and 97.5th percentiles using the DRE method,
averaged over the 100 simulations, and E2.5 and E97.5 represent these same averages
using the samples’ empirical CDFs. For the Weibull quantile estimation, we esti-
mated via maximum likelihood estimation an estimator for τ assuming the data were
a random sample from a generalized gamma distribution. We then fixed this esti-
mate as the value for τ in the density ratio and proceeded with the semiparametric
estimation.
We notice that for all of the sample sizes and distributions examined, MSEDRE2.5
andMSEDRE97.5 are substantially lower thanMSEE2.5 andMSEE97.5 . Thus, by sim-
ply fusing samples from known distributions with uniformly distributed data and
estimating the sample’s distribution using the semiparametric DRE approach with
the combined data based on a reasonable choice of a tilt function, we are able to more
accurately determine the 2.5th and 97.5th percentiles. As our analysis illustrates,
for data that appears to be normally distributed (i.e. symmetric and unimodal, the
tilt function h(x) = (x, x2)′ is reasonable, whereas for skewed data a tilt function of
the form h(x) = (x, log(x))′ or more generally h(x) = (xτ , log(x))′ is acceptable.
83
Tab. 4.1: Simulation Results based on Samples from a Normal Distribution with mean
µ and standard deviation σ, fused with uniformly distributed sample over xmin
and xmax, assuming tilt function h(x) = (x, x2)′
N µ σ MSEDRE2.5 MSEDRE97.5 MSEE2.5 MSEE97.5
600 0 1 0.006 0.007 0.012 0.011
600 10 2 0.030 0.018 0.041 0.042
600 5 3 0.076 0.084 0.143 0.111
600 -2 5 0.176 0.190 0.234 0.222
1000 0 1 0.004 0.005 0.008 0.008
1000 10 2 0.015 0.020 0.021 0.027
1000 5 3 0.046 0.040 0.070 0.066
1000 -2 5 0.101 0.119 0.164 0.204
5000 0 1 0.001 0.001 0.001 0.002
5000 10 2 0.018 0.015 0.027 0.031
5000 5 3 0.039 0.041 0.056 0.066
5000 -2 5 0.103 0.114 0.159 0.184
10000 0 1 < 0.001 0.001 0.001 0.001
10000 10 2 0.017 0.017 0.034 0.029
10000 5 3 0.038 0.044 0.060 0.068
10000 -2 5 0.118 0.118 0.164 0.184
15000 0 1 < 0.001 < 0.001 0.001 < 0.001
15000 10 2 0.001 0.001 0.002 0.002
15000 5 3 0.003 0.003 0.004 0.005
15000 -2 5 0.008 0.007 0.012 0.010
20000 0 1 < 0.001 < 0.001 < 0.001 < 0.001
20000 10 2 0.017 0.014 0.029 0.029
20000 5 3 0.041 0.031 0.075 0.070
20000 -2 5 0.090 0.116 0.195 0.161
25000 0 1 < 0.001 < 0.001 < 0.001 < 0.001
25000 10 2 0.017 0.017 0.035 0.024
25000 5 3 0.051 0.042 0.066 0.079
25000 -2 5 0.113 0.067 0.234 0.143
84
Tab. 4.2: Further simulations based on Samples from a student-t Distribution with varying
degrees of freedom, fused with uniformly distributed sample over xmin and xmax,
assuming tilt function h(x) = (x, x2)′
N df MSEDRE2.5 MSEDRE97.5 MSEE2.5 MSEE97.5
600 10 0.0127 0.0139 0.0253 0.0234
600 15 0.0114 0.0094 0.0191 0.0190
600 20 0.0089 0.0110 0.0146 0.0165
600 25 0.0101 0.0107 0.0147 0.0157
600 30 0.0076 0.0113 0.0140 0.0200
600 35 0.0069 0.0070 0.0118 0.0140
600 40 0.0087 0.0079 0.0148 0.0128
1000 10 0.0127 0.0139 0.0253 0.0234
1000 15 0.0114 0.0094 0.0191 0.0190
1000 20 0.0089 0.0110 0.0146 0.0165
1000 25 0.0101 0.0107 0.0147 0.0157
1000 30 0.0076 0.0113 0.0140 0.0200
1000 35 0.0069 0.0070 0.0118 0.0140
1000 40 0.0087 0.0079 0.0148 0.0128
5000 10 0.0020 0.0017 0.0026 0.0027
5000 15 0.0014 0.0015 0.0024 0.0021
5000 20 0.0015 0.0013 0.0024 0.0023
5000 25 0.0012 0.0013 0.0023 0.0022
5000 30 0.0010 0.0010 0.0016 0.0016
5000 35 0.0011 0.0018 0.0019 0.0016
5000 40 0.0010 0.0010 0.0020 0.0017
10000 10 0.0018 0.0019 0.0012 0.0011
10000 15 0.0017 0.0016 0.0012 0.0010
10000 20 0.0016 0.0016 0.0019 0.0011
10000 25 0.0015 0.0014 0.0019 0.0017
10000 30 0.0016 0.0016 0.0017 0.0010
10000 35 0.0015 0.0014 0.0017 0.0017
10000 40 0.0015 0.0015 0.0018 0.0017
85
Tab. 4.3: Further simulations based on Samples from a student-t Distribution with varying
degrees of freedom, fused with uniformly distributed sample over xmin and xmax,
assuming tilt function h(x) = (x, x2)′
N df MSEDRE2.5 MSEDRE97.5 MSEE2.5 MSEE97.5
15000 10 0.0017 0.0017 0.0019 0.0019
15000 15 0.0015 0.0015 0.0018 0.0017
15000 20 0.0014 0.0014 0.0017 0.0015
15000 25 0.0014 0.0014 0.0018 0.0016
15000 30 0.0014 0.0014 0.0015 0.0015
15000 35 0.0014 0.0013 0.0016 0.0015
15000 40 0.0014 0.0014 0.0016 0.0016
20000 10 0.0018 0.0016 0.0019 0.0017
20000 15 0.0014 0.0013 0.0015 0.0015
20000 20 0.0012 0.0013 0.0015 0.0015
20000 25 0.0012 0.0013 0.0013 0.0014
20000 30 0.0013 0.0013 0.0014 0.0015
20000 35 0.0012 0.0013 0.0013 0.0015
20000 40 0.0012 0.0012 0.0013 0.0013
25000 10 0.0016 0.0015 0.0016 0.0016
25000 15 0.0013 0.0013 0.0014 0.0014
25000 20 0.0012 0.0013 0.0013 0.0014
25000 25 0.0012 0.0012 0.0014 0.0014
25000 30 0.0012 0.0013 0.0013 0.0015
25000 35 0.0012 0.0012 0.0014 0.0014
25000 40 0.0012 0.0013 0.0014 0.0014
86
Tab. 4.4: Simulations based on Samples from a Gamma Distribution with parameters α
and β, assuming tilt function h(x) = (x, log(x))′
N α β MSEDRE2.5 MSEDRE97.5 MSEE2.5 MSEE97.5
600 1 2 0.001 1.367 0.001 2.040
600 3 2 0.092 2.170 0.116 2.655
600 5 2 0.235 2.898 0.279 3.644
600 1 4 < 0.001 0.297 < 0.001 0.366
600 3 4 0.030 0.495 0.032 0.601
600 5 4 0.065 0.642 0.078 0.979
600 7 4 0.104 0.898 0.122 1.073
600 1 8 < 0.001 0.083 < 0.001 0.095
600 3 8 0.005 0.149 0.005 0.167
600 5 8 0.021 0.240 0.024 0.254
600 7 8 0.030 0.255 0.037 0.304
1000 1 2 0.001 0.671 0.001 1.006
1000 3 2 0.053 0.930 0.058 1.287
1000 5 2 0.167 1.693 0.248 2.331
1000 1 4 < 0.001 0.174 < 0.001 0.232
1000 3 4 0.011 0.324 0.013 0.427
1000 5 4 0.046 0.339 0.049 0.467
1000 7 4 0.068 0.559 0.083 0.896
1000 1 8 < 0.001 0.066 < 0.001 0.070
1000 3 8 0.003 0.085 0.003 0.112
1000 5 8 0.013 0.116 0.015 0.138
1000 7 8 0.016 0.148 0.021 0.191
87
Tab. 4.5: Further simulations based on Samples from a Gamma Distribution with param-
eters α and β, assuming tilt function h(x) = (x, log(x))′
N α β MSEDRE2.5 MSEDRE97.5 MSEE2.5 MSEE97.5
5000 1 2 < 0.001 0.133 < 0.001 0.211
5000 3 2 0.010 0.214 0.012 0.251
5000 5 2 0.035 0.251 0.041 0.423
5000 1 4 < 0.001 0.042 < 0.001 0.053
5000 3 4 0.003 0.070 0.003 0.087
5000 5 4 0.009 0.093 0.011 0.107
5000 7 4 0.014 0.113 0.018 0.148
5000 1 8 < 0.001 0.010 < 0.001 0.013
5000 3 8 0.001 0.021 0.001 0.022
5000 5 8 0.002 0.024 0.002 0.026
5000 7 8 0.004 0.021 0.004 0.034
10000 1 2 < 0.001 0.074 < 0.001 0.098
10000 3 2 0.004 0.110 0.005 0.161
10000 5 2 0.013 0.160 0.016 0.257
10000 1 4 < 0.001 0.019 < 0.001 0.025
10000 3 4 0.002 0.041 0.002 0.052
10000 5 4 0.004 0.037 0.005 0.047
10000 7 4 0.008 0.051 0.009 0.071
10000 1 8 < 0.001 0.005 < 0.001 0.006
10000 3 8 < 0.001 0.008 < 0.001 0.010
10000 5 8 0.001 0.012 0.001 0.014
10000 7 8 0.002 0.014 0.002 0.016
15000 1 2 < 0.001 0.047 < 0.001 0.063
15000 3 2 0.004 0.070 0.005 0.084
15000 5 2 0.011 0.107 0.013 0.161
15000 1 4 < 0.001 0.015 < 0.001 0.017
15000 3 4 0.001 0.019 0.001 0.023
15000 5 4 0.003 0.036 0.004 0.045
15000 7 4 0.005 0.034 0.006 0.048
15000 1 8 < 0.001 0.004 < 0.001 0.005
15000 3 8 < 0.001 0.006 < 0.001 0.007
15000 5 8 0.001 0.008 0.001 0.009
15000 7 8 0.001 0.010 0.001 0.011
88
Tab. 4.6: Further simulations based on Samples from a Gamma Distribution with param-
eters α and β, assuming tilt function h(x) = (x, log(x))′
N α β MSEDRE2.5 MSEDRE97.5 MSEE2.5 MSEE97.5
20000 1 2 < 0.001 0.047 < 0.001 0.065
20000 3 2 0.003 0.055 0.003 0.079
20000 5 2 0.007 0.088 0.009 0.123
20000 1 4 < 0.001 0.010 < 0.001 0.011
20000 3 4 0.001 0.023 0.001 0.026
20000 5 4 0.001 0.020 0.002 0.024
20000 7 4 0.004 0.024 0.004 0.033
20000 1 8 < 0.001 0.002 < 0.001 0.002
20000 3 8 < 0.001 0.004 < 0.001 0.005
20000 5 8 0.001 0.006 0.001 0.007
20000 7 8 0.001 0.007 0.001 0.008
25000 1 2 < 0.001 0.027 < 0.001 0.035
25000 3 2 0.002 0.051 0.003 0.068
25000 5 2 0.007 0.082 0.009 0.133
25000 1 4 < 0.001 0.006 < 0.001 0.007
25000 3 4 0.001 0.013 0.001 0.018
25000 5 4 0.001 0.016 0.002 0.026
25000 7 4 0.002 0.018 0.003 0.024
25000 1 8 < 0.001 0.002 < 0.001 0.002
25000 3 8 < 0.001 0.003 < 0.001 0.004
25000 5 8 < 0.001 0.005 0.001 0.005
25000 7 8 0.001 0.007 0.001 0.009
89
Tab. 4.7: Simulations based on Samples from a Weibull Distribution with parameters α
and β, assuming tilt function h(x) = (xτ , log(x))′
N α β MSEDRE2.5 MSEDRE97.5 MSEE2.5 MSEE97.5
600 1 1 0.004 6.345 0.005 6.585
600 1.2 1 0.009 2.726 0.011 3.660
600 1.5 1 0.016 0.955 0.021 1.134
600 2.5 1 0.052 0.140 0.063 0.201
600 1 1.2 0.005 6.388 0.005 7.254
600 1.2 1.2 0.015 2.450 0.016 2.679
600 1.5 1.2 0.024 1.150 0.029 1.790
600 2.5 1.2 0.079 0.212 0.099 0.266
600 1 1.5 0.010 10.454 0.011 10.995
600 1.2 1.5 0.022 4.000 0.028 5.585
600 1.5 1.5 0.049 2.727 0.064 2.945
600 2.5 1.5 0.087 0.364 0.118 0.491
600 1 2.5 0.033 29.605 0.035 39.361
600 1.2 2.5 0.068 12.825 0.072 16.565
600 1.5 2.5 0.124 5.034 0.142 8.541
600 2.5 2.5 0.274 1.023 0.370 1.202
1000 1 1 0.003 2.524 0.003 3.272
1000 1.2 1 0.006 1.417 0.008 1.653
1000 1.5 1 0.010 0.551 0.013 0.722
1000 2.5 1 0.021 0.079 0.033 0.119
1000 1 1.2 0.004 3.900 0.005 5.232
1000 1.2 1.2 0.009 1.838 0.009 2.516
1000 1.5 1.2 0.015 0.727 0.019 1.008
1000 2.5 1.2 0.034 0.125 0.041 0.129
1000 1 1.5 0.006 7.696 0.006 9.494
1000 1.2 1.5 0.015 2.571 0.017 3.905
1000 1.5 1.5 0.025 1.546 0.033 1.871
1000 2.5 1.5 0.068 0.181 0.084 0.232
1000 1 2.5 0.012 15.829 0.012 22.411
1000 1.2 2.5 0.041 8.823 0.048 12.399
1000 1.5 2.5 0.081 3.573 0.101 4.808
1000 2.5 2.5 0.198 0.662 0.236 0.755
90
Tab. 4.8: Further Simulations based on Samples from a Weibull Distribution with param-
eters α and β, assuming tilt function h(x) = (xτ , log(x))′
N α β MSEDRE2.5 MSEDRE97.5 MSEE2.5 MSEE97.5
5000 1 1 < 0.001 0.459 < 0.001 0.559
5000 1.2 1 0.001 0.256 0.001 0.353
5000 1.5 1 0.003 0.097 0.003 0.122
5000 2.5 1 0.005 0.021 0.007 0.025
5000 1 1.2 0.001 0.931 0.001 1.274
5000 1.2 1.2 0.002 0.452 0.002 0.722
5000 1.5 1.2 0.004 0.184 0.004 0.253
5000 2.5 1.2 0.008 0.029 0.009 0.036
5000 1 1.5 0.002 1.319 0.002 2.066
5000 1.2 1.5 0.002 0.452 0.002 0.722
5000 1.5 1.5 0.004 0.280 0.004 0.351
5000 2.5 1.5 0.011 0.035 0.016 0.037
5000 1 2.5 0.003 0.626 0.003 0.759
5000 1.2 2.5 0.007 1.608 0.008 1.826
5000 1.5 2.5 0.010 0.585 0.016 0.693
5000 2.5 2.5 0.045 0.116 0.054 0.169
4.3 The Impact of Welfare Reform
4.3.1 Data
We obtained data of AFDC (1982-1996) monthly caseloads and annual dis-
continuances in caseloads due to employment from the Quarterly Public Assistance
Statistics [157]. The TANF data (1997-2009) was obtained from the HHS website
as well as the each of program’s annual reports to Congress [25–33]. Because the
data consisted of thousands of caseloads and discontinuances for each each state,
we changed the units of our data to thousands of caseloads to make our estima-
tion easier. Additionally, we divided the annual discontinuance by 12 to represent
average monthly discontinuances to have both covariates have the same units of
measurement. Across all 53 territories, AFDC programs had slightly more than
75,000 caseloads with approximately 7,500 discontinuances due to employment over
91
Tab. 4.9: Further simulations based on Samples from a Weibull Distribution with param-
eters α and β, assuming tilt function h(x) = (xτ , log(x))′
N α β MSEDRE2.5 MSEDRE97.5 MSEE2.5 MSEE97.5
10000 1 1 < 0.001 0.319 < 0.001 0.392
10000 1.2 1 < 0.001 0.124 0.001 0.154
10000 1.5 1 0.001 0.059 0.001 0.063
10000 2.5 1 0.003 0.008 0.003 0.009
10000 1 1.2 0.002 3.210 0.002 4.069
10000 1.2 1.2 0.001 0.186 0.001 0.267
10000 1.5 1.2 0.002 0.092 0.002 0.113
10000 2.5 1.2 0.004 0.014 0.005 0.020
10000 1 1.5 0.001 0.634 0.001 0.761
10000 1.2 1.5 0.001 0.339 0.001 0.456
10000 1.5 1.5 0.003 0.095 0.003 0.152
10000 2.5 1.5 0.005 0.024 0.006 0.029
10000 1 2.5 0.002 1.924 0.002 2.693
10000 1.2 2.5 0.003 0.733 0.004 1.172
10000 1.5 2.5 0.006 0.276 0.008 0.361
10000 2.5 2.5 0.016 0.071 0.018 0.090
15000 1 1 < 0.001 0.222 < 0.001 0.284
15000 1.2 1 < 0.001 0.106 < 0.001 0.134
15000 1.5 1 0.001 0.034 0.001 0.043
15000 2.5 1 0.002 0.009 0.003 0.010
15000 1 1.2 < 0.001 0.282 < 0.001 0.476
15000 1.2 1.2 < 0.001 0.108 0.001 0.154
15000 1.5 1.2 0.001 0.049 0.001 0.077
15000 2.5 1.2 0.003 0.009 0.003 0.012
15000 1 1.5 < 0.001 0.377 < 0.001 0.515
15000 1.2 1.5 0.001 0.196 0.001 0.265
15000 1.5 1.5 0.002 0.082 0.002 0.108
15000 2.5 1.5 0.004 0.014 0.005 0.020
15000 1 2.5 0.001 1.260 0.001 1.896
15000 1.2 2.5 0.002 0.432 0.002 0.633
15000 1.5 2.5 0.005 0.197 0.006 0.304
15000 2.5 2.5 0.009 0.039 0.011 0.075
92
Tab. 4.10: Further simulations based on Samples from a Weibull Distribution with pa-
rameters α and β, assuming tilt function h(x) = (xτ , log(x))′
N α β MSEDRE2.5 MSEDRE97.5 MSEE2.5 MSEE97.5
20000 1 1 < 0.001 0.175 < 0.001 0.201
20000 1.2 1 < 0.001 0.059 < 0.001 0.075
20000 1.5 1 0.001 0.024 0.001 0.032
20000 2.5 1 0.001 0.006 0.001 0.007
20000 1 1.2 < 0.001 0.307 < 0.001 0.348
20000 1.2 1.2 0.001 0.100 0.001 0.124
20000 1.5 1.2 0.001 0.039 0.001 0.061
20000 2.5 1.2 0.002 0.009 0.003 0.011
20000 1 1.5 < 0.001 0.307 < 0.001 0.442
20000 1.2 1.5 0.001 0.129 0.001 0.186
20000 1.5 1.5 0.001 0.061 0.001 0.072
20000 2.5 1.5 0.003 0.014 0.003 0.015
20000 1 2.5 0.001 0.861 0.001 1.124
20000 1.2 2.5 0.002 0.377 0.002 0.505
20000 1.5 2.5 0.004 0.191 0.004 0.265
20000 2.5 2.5 0.008 0.030 0.009 0.041
25000 1 1 < 0.001 0.125 < 0.001 0.150
25000 1.2 1 < 0.001 0.045 < 0.001 0.065
25000 1.5 1 0.001 0.019 0.001 0.025
25000 2.5 1 0.001 0.004 0.001 0.005
25000 1 1.2 < 0.001 0.146 < 0.001 0.208
25000 1.2 1.2 < 0.001 0.089 < 0.001 0.118
25000 1.5 1.2 0.001 0.030 0.001 0.035
25000 2.5 1.2 0.001 0.007 0.002 0.008
25000 1 1.5 < 0.001 0.170 < 0.001 0.277
25000 1.2 1.5 < 0.001 0.103 < 0.001 0.137
25000 1.5 1.5 0.001 0.045 0.001 0.063
25000 2.5 1.5 0.003 0.009 0.003 0.010
25000 1 2.5 0.001 0.946 0.001 1.090
25000 1.2 2.5 0.001 0.403 0.001 0.482
25000 1.5 2.5 0.003 0.118 0.004 0.181
25000 2.5 2.5 0.007 0.027 0.009 0.044
93
a fifteen year time horizon. TANF, on the other hand, had over 40,000 caseloads
with over 7,000 discontinuances due to employment over the course of thirteen years.
4.3.2 Analysis of the PROWA Using Bayesian Lagged Regression
Using our data, we regressed our dependent variables, yt discontinuances in
caseloads due to employment at year t, against our explanatory variable, total
caseloads in a previous year t′, xt′ . We estimated ten instances of the model defined
in equation 3. In particular, we estimated separate regressions regarding the impact
of each of the five previous year’s caseloads (t − 1, t − 2, t − 3, t − 4, and t − 5)
on discontinuances due to employment for [1] AFDC programs and for [2] TANF
programs as TANF had instituted a five year time limit.2
We used WinBUGS to run a Bayesian MCMC sampler over the parameter
space of the hierarchical model outlined in equations (4.1) to (4.8) to sample the
posterior distribution [50, 97]. We ran our MCMC sampler for 30,000 draws, using
the first half for “burn-in” to dissipate initial conditions. In the above model, Φ0
and ν0 of equation 6 were randomly generated. We ran our simulations several
times over different random choices of Φ0 and ν0 and found similar results. These
results suggested that the random generation of Φ0 and ν0 had no impact on the
results generated by our MCMC sampler. Additionally, our posterior samples overall
had a low degree of autocorrelation, indicating that we had likely not oversampled
particular regions within our density.
After completion of the MCMC sampler, we estimated posterior means, stan-
dard deviations, and 95% credible intervals of all posterior samples.3 Since we had
truncated the MCMC sampler after 30,000 draws, however, there is still some con-
2 Concatenating the explanatory variables across multiple lags for a larger multiple regression
analysis resulted in multiciollinearity issues due to correlations between lagged caseloads and hence
was not performed as part of this analysis.
3 A 95% credible interval for a univariate sample is a Bayesian interval estimator and denotes
an interval bounded by the 2.5th and 97.5th percentiles of the posterior sample, serving as an
estimate for the interval bounded by the 2.5th and 97.5th quantiles of the distribution.
94
Fig. 4.1: Histogram of Posterior Sample for Posterior Intercept Mean Coefficient (µ0),
TANF regression, Lag 1
cern about missing information in the posterior sample. We therefore used the DRE
method to improve upon our posterior sample. The posterior densities for µ0 and
µ1 were symmetric and unimodal and were therefore appropriate for analysis using
the tilt function h(x) = (x, x2)′ described in (4.12). The posterior densities of σ20
and σ21, however, appeared to be slightly skewed and we therefore more appropriate
for analysis using the tilt function h(x) = (x, log(x))′ described in (4.13). The plots
for a few selected coefficients are depicted in Figures 4.1-4.4 illustrating the reason
for our choices of tilt functions:
For our posterior samples of µ0 and µ1, we took each sample of the marginal
posterior (sample size 15,000), used as a reference density a random sample of equal
size from a uniform distribution over the posterior sample’s range, and applied
the DRE method to estimate the cumulative distribution function (CDF) of the
marginal posterior for this new sample of size 30,000. We then equated the CDF to
95
Fig. 4.2: Histogram of Posterior Slope Mean Coefficient (µ1), TANF regression, Lag 1
Fig. 4.3: Histogram of Posterior Intercept Variance Coefficient (σ20), TANF regression,
Lag 1
96
Fig. 4.4: Posterior Slope Variance Coefficient, TANF regression (σ21), Lag 1
0.025 and solved the resulting equation to generate a lower limit for the 95% credible
interval. We took a similar approach, equating the CDF to 0.975, to generate an
upper estimate for the credible interval. We refer to this newly estimated region,
based on semiparametric out of sample fusion, as the posterior density’s 95% credible
interval. Our results are outlined in Tables 4.11-4.30.
Tab. 4.11: AFDC Regression of Discontinuances as a Function of Caseloads - One Year
Time Lag
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
µ0 0.072 0.173 -0.263 0.417
µ1 0.009 0.075 -0.136 0.155
σ20 0.773 0.188 0.486 1.227
σ21 0.290 0.049 0.208 0.400
97
Tab. 4.12: AFDC Regression of Discontinuances as a Function of Caseloads, Interval Es-
timates and Refinements - One Year Time Lag
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
µ0 -0.263 0.417 -0.264 0.411
µ1 -0.136 0.155 -0.136 0.156
σ20 0.486 1.227 0.471 1.192
σ21 0.208 0.400 0.204 0.396
Tab. 4.13: TANF Regression of Discontinuances as a Function of Caseloads - One Year
Time Lag
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
µ0 0.068 0.151 -0.228 0.365
µ1 0.014 0.075 -0.132 0.163
σ20 0.630 0.142 0.408 0.961
σ21 0.292 0.050 0.209 0.406
Tab. 4.14: TANF Regression of Discontinuances as a Function of Caseloads, Interval Es-
timates and Refinements - One Year Time Lag
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
µ0 -0.228 0.365 -0.228 0.364
µ1 -0.132 0.163 -0.135 0.162
σ20 0.408 0.961 0.398 0.947
σ21 0.209 0.406 0.206 0.401
Tab. 4.15: AFDC Regression of Discontinuances as a Function of Caseloads - Two Year
Time Lag
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
µ0 0.272 0.183 -0.094 0.624
µ1 0.001 0.074 -0.146 0.149
σ20 0.761 0.186 0.479 1.196
σ21 0.290 0.050 0.208 0.405
98
Tab. 4.16: AFDC Regression of Discontinuances as a Function of Caseloads, Interval Es-
timates and Refinements - Two Year Time Lag
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
µ0 -0.094 0.624 -0.088 0.628
µ1 -0.146 0.149 -0.145 0.148
σ20 0.479 1.196 0.462 1.172
σ21 0.208 0.405 0.204 0.399
Tab. 4.17: TANF Regression of Discontinuances as a Function of Caseloads - Two Year
Time Lag
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
µ0 0.240 0.151 -0.059 0.535
µ1 0.000 0.076 -0.148 0.150
σ20 0.578 0.125 0.380 0.870
σ21 0.292 0.050 0.210 0.404
Tab. 4.18: TANF Regression of Discontinuances as a Function of Caseloads, Interval Es-
timates and Refinements - Two Year Time Lag
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
µ0 -0.059 0.535 -0.059 0.534
µ1 -0.148 0.150 -0.149 0.150
σ20 0.380 0.870 0.372 0.854
σ21 0.210 0.404 0.207 0.399
4.3.3 Statistical Inferences Based on Hierarchical Bayesian Analysis of PROWA
Although our overall coefficient estimates are small in magnitude, the reader
should be reminded of the fact that the data set analyzed consisted of observations
in units of thousands. Therefore, slight differences between the coefficients signify
notable differences. For the first year time lag model, the marginal posterior mean
for µ1 is quite higher in Table 4.11 than in Table 4.13. A two sample Kolmogrov
Smirnov test of the sample of the two posterior samples rejects the null hypothesis
of distributional equality at p < 0.001, suggesting that there is a marked difference
99
Tab. 4.19: AFDC Regression of Discontinuances as a Function of Caseloads - Three Year
Time Lag
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
µ0 0.196 0.196 -0.184 0.582
µ1 0.000 0.076 -0.147 0.150
σ20 0.765 0.201 0.465 1.237
σ21 0.292 0.050 0.210 0.406
Tab. 4.20: AFDC Regression of Discontinuances as a Function of Caseloads, Interval Es-
timates and Refinements - Three Year Time Lag
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
µ0 -0.184 0.582 -0.189 0.583
µ1 -0.147 0.150 -0.148 0.150
σ20 0.465 1.237 0.448 1.212
σ21 0.210 0.406 0.206 0.401
Tab. 4.21: TANF Regression of Discontinuances as a Function of Caseloads - Three Year
Time Lag
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
µ0 0.390 0.167 0.069 0.719
µ1 -0.005 0.075 -0.153 0.143
σ20 0.716 0.172 0.449 1.121
σ21 0.291 0.050 0.208 0.405
Tab. 4.22: TANF Regression of Discontinuances as a Function of Caseloads, Interval Es-
timates and Refinements - Three Year Time Lag
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
µ0 0.069 0.719 0.065 0.720
µ1 -0.153 0.143 -0.154 0.143
σ20 0.449 1.121 0.437 1.095
σ21 0.208 0.405 0.204 0.399
100
Tab. 4.23: AFDC Regression of Discontinuances as a Function of Caseloads - Four Year
Time Lag
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
µ0 0.149 0.212 -0.26 0.563
µ1 0.003 0.076 -0.147 0.153
σ20 0.722 0.175 0.453 1.13
σ21 0.292 0.05 0.211 0.407
Tab. 4.24: AFDC Regression of Discontinuances as a Function of Caseloads, Interval Es-
timates and Refinements - Four Year Time Lag
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
µ0 -0.26 0.563 -0.265 0.562
µ1 -0.147 0.153 -0.146 0.152
σ20 0.453 1.13 0.441 1.108
σ21 0.211 0.407 0.206 0.401
Tab. 4.25: TANF Regression of Discontinuances as a Function of Caseloads - Four Year
Time Lag
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
µ0 0.44 0.171 0.111 0.788
µ1 -0.008 0.075 -0.156 0.141
σ20 0.821 0.194 0.513 1.266
σ21 0.293 0.051 0.21 0.407
Tab. 4.26: TANF Regression of Discontinuances as a Function of Caseloads, Interval Es-
timates and Refinements - Four Year Time Lag
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
µ0 0.111 0.788 0.106 0.782
µ1 -0.156 0.141 -0.156 0.141
σ20 0.513 1.266 0.5 1.246
σ21 0.21 0.407 0.206 0.403
101
Tab. 4.27: AFDC Regression of Discontinuances as a Function of Caseloads - Five Year
Time Lag
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
µ0 0.269 0.241 -0.203 0.742
µ1 -0.002 0.076 -0.15 0.145
σ20 0.822 0.213 0.501 1.325
σ21 0.292 0.051 0.21 0.407
Tab. 4.28: AFDC Regression of Discontinuances as a Function of Caseloads, Interval Es-
timates and Refinements - Five Year Time Lag
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
µ0 -0.203 0.742 -0.207 0.74
µ1 -0.15 0.145 -0.15 0.145
σ20 0.501 1.325 0.483 1.296
σ21 0.21 0.407 0.206 0.402
Tab. 4.29: TANF Regression of Discontinuances as a Function of Caseloads - Five Year
Time Lag
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
µ0 0.541 0.175 0.205 0.889
µ1 -0.013 0.075 -0.159 0.138
σ20 0.744 0.191 0.458 1.19
σ21 0.292 0.05 0.21 0.404
Tab. 4.30: TANF Regression of Discontinuances as a Function of Caseloads, Interval Es-
timates and Refinements - Five Year Time Lag
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
µ0 0.205 0.889 0.198 0.886
µ1 -0.159 0.138 -0.161 0.136
σ20 0.458 1.19 0.444 1.171
σ21 0.21 0.404 0.207 0.401
102
between the two posterior distributions.4
Overall, these result suggests that the previous year’s caseloads for TANF had
a notably stronger impact in getting people off the welfare rolls and back to work
than the previous year’s caseloads for AFDC programs did. The other models with
the longer lagged covariates suggest that total caseloads had either a negligible or
even a negative effect on employment-based discontinuances.
There are also significant differences in using DRE to refine the 95% credible
intervals. For example, the interval estimates regarding µ1 are altered by a factor of
several thousand discontinuances per caseload, depending on the year. Additionally,
the posterior distributions of the variances are also altered substantially. Regardless,
however, the posterior estimates of σ20 and σ
2
1 all remain relatively small, suggesting
that there is not much uncertainty pertaining to the relationship between caseloads
and discontinuances.
An additional fully frequentist statistical analysis is included in this chapter’s
Appendix. This analysis also finds similar results to those above.
4.4 Additional Analysis
In addition to the analysis above, we conducted further analysis to examine
the efficacy of transforming AFDC to TANF. As our data of caseloads and discon-
tinuances spanned from 1982-2009, we conducted another rigorous Bayesian analysis
over the entire time horizon, with categorically-coded coefficients representing time
in terms of year. Comparisons of these coefficient estimates from before the PROWA
4 As mentioned earlier, autocorrelations of the sample were quite low, suggesting that indepen-
dence of the sample, a sufficient condition for being able to apply the Kolmogorov Smirnov test,
was not an unreasonable assumption.
103
to after the PROWA can enable us to understand the efficacy of the reforms them-
selves.
Again, following the notation corresponding to our previous model, we can
again define this new hierarchical Bayesian model over i ∈ {1, . . . , 53} territories
(50 states as well as Guam, Puerto Rico, and the District of Columbia) (units) over
t ∈ {1982, . . . , 2008} years as follows:
yi ∼MVN(βi,0 + βi,1xi,1 + βt,Σ) (4.25)
βi,0 ∼ N(µ0, σ
2
0) (4.26)
βi,1 ∼ N(µ1, σ
2
1) (4.27)
βt ∼ N(0, 10) (4.28)
Σ−1 ∼Wishart(Φ0, ν0) (4.29)
µ0 ∼ N(0, 10) (4.30)
µ1 ∼ N(0, 10) (4.31)
σ−20 ∼ Γ(10, 10) (4.32)
σ−21 ∼ Γ(10, 10) (4.33)
This model is quite similar to the model we examined earlier, but runs across
the data set’s entire time horizon and also contains a term βt that quantifies the
impact of time t′ on caseload reduction at time t. A posterior examination of this
coefficient before and after the introduction of TANF can enable us to understand
the impact of the 1996 reform. Just as we did earlier, we defined β1,0 = 0 as well
as β1982=0 to ensure statistical identifiability of our model. We ran this Bayesian
MCMC sampler for 30,000 iterations, using first half to dissipate our initial con-
ditions and the remaining half for statistical inference. Our Bayesian posterior
104
estimates are outlined in Tables 4.31-4.32. As our posterior samples of µ0, µ1, and
βt appeared to be symmetric and unimodal we again utilized our DRE approach to
improve the estimation of our Bayesian credible intervals.
Tab. 4.31: Additional Analysis
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
µ0 0.093 0.167 -0.231 0.428
µ1 0.006 0.074 -0.139 0.150
β1983 0.042 0.067 -0.091 0.174
β1984 -0.008 0.062 -0.131 0.113
β1985 -0.028 0.073 -0.175 0.114
β1986 0.010 0.080 -0.146 0.165
β1987 0.011 0.077 -0.142 0.160
β1988 0.032 0.082 -0.130 0.191
β1989 0.015 0.081 -0.146 0.175
β1990 0.048 0.082 -0.116 0.206
β1991 0.086 0.087 -0.091 0.256
β1992 0.076 0.082 -0.091 0.232
β1993 0.064 0.091 -0.120 0.240
β1994 0.199 0.100 0.005 0.394
β1995 0.189 0.086 0.019 0.356
β1996 0.573 0.183 0.201 0.919
β1997 0.512 0.158 0.191 0.811
β1998 0.290 0.123 0.044 0.531
β1999 0.294 0.111 0.074 0.511
β2000 0.249 0.104 0.040 0.451
β2002 0.272 0.115 0.048 0.498
β2003 0.305 0.116 0.079 0.534
β2004 0.294 0.115 0.069 0.518
β2005 0.344 0.134 0.084 0.613
β2006 0.382 0.162 0.060 0.702
β2007 0.298 0.151 -0.003 0.589
β2008 0.256 0.157 -0.059 0.568
σ20 0.778 0.164 0.508 1.151
σ21 0.288 0.049 0.207 0.397
We see a noticeable increase in our coefficient estimates pertaining to time
after 1997, once the law had become fully enacted. In fact, the small posterior
estimates of βt from the 1980s and early 1990s suggest that AFDC programs at the
105
Tab. 4.32: Additional Analysis, Interval Estimates and Refinements
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Lower Limit Upper Limit
µ0 -0.231 0.428 -0.234 0.427
µ1 -0.139 0.150 -0.138 0.151
β1983 -0.091 0.174 -0.091 0.173
β1984 -0.131 0.113 -0.130 0.113
β1985 -0.175 0.114 -0.173 0.116
β1986 -0.146 0.165 -0.148 0.166
β1987 -0.142 0.160 -0.140 0.161
β1988 -0.130 0.191 -0.129 0.193
β1989 -0.146 0.175 -0.145 0.175
β1990 -0.116 0.206 -0.114 0.208
β1991 -0.091 0.256 -0.089 0.256
β1992 -0.091 0.232 -0.087 0.235
β1993 -0.120 0.240 -0.117 0.241
β1994 0.005 0.394 0.004 0.393
β1995 0.019 0.356 0.019 0.357
β1996 0.201 0.919 0.210 0.926
β1997 0.191 0.811 0.198 0.818
β1998 0.044 0.531 0.046 0.532
β1999 0.074 0.511 0.074 0.513
β2000 0.040 0.451 0.042 0.453
β2002 0.048 0.498 0.047 0.496
β2003 0.079 0.534 0.076 0.535
β2004 0.069 0.518 0.069 0.518
β2005 0.084 0.613 0.082 0.608
β2006 0.060 0.702 0.061 0.700
β2007 -0.003 0.589 0.001 0.590
β2008 -0.059 0.568 -0.053 0.567
σ20 0.508 1.151 0.500 1.134
σ21 0.207 0.397 0.203 0.392
time were essentially just a poverty trap, lulling people into poverty and hindering
upward mobility. These posterior estimates increase notably in the 1990s, especially
after the law was introduced in 1996.
In Table 32, we present our Bayesian 95% credible intervals and subsequent
refinements. We find that all 95% credible intervals regarding time before the 1996
welfare reform all had negative probability mass, whereas after the law had become
fully enacted, they had no negative mass until β2007 and β2008. After our posterior
refinements, however, all were positive except for β2008. This negativity was due to
our MCMC sampler not properly summarizing the posterior. The higher point and
interval estimates of these “post-welfare reform” coefficients, indicate that the law
had indeed been successful in reducing dependence on government assistance.
4.4.1 Policy Implications
Our analysis clearly illustrates the great success of welfare reform. Specifically,
our Bayesian models suggest that the previous year’s caseloads had a significantly
106
more notable impact on getting people off of the welfare rolls and back to work
under the TANF program compared to the AFDC program, particularly within one
year. Subsequent year lags had a negligible or even a detrimental impact on caseload
reductions due to employment.
These general findings are substantiated by research produced by The Her-
itage Foundation, the Cato Institute, and the Brookings Institution. In particular,
research from these well-known think tanks has clearly illustrated that as a re-
sult of welfare reform, caseloads declined, earnings increased, and poverty rates
dropped [61, 128, 145, 152]. More rigorous work standards and time limits would
almost surely improve on these results.
4.5 Conclusions and Future Research
Our study offers a number of significant contributions. Firstly, our work also
sheds light on the efficacy of welfare reform. Although safety nets are an impor-
tant component of society, they should be constructed in a manner to help provide
temporary assistance to get people back to work and become contributing members
of society. If these programs are just an open-ended government handout, recipi-
ents can become overly dependent on them and can subsequently become incapable
of realizing the American dream. As Charles Murray’s Losing Ground illustrated,
these programs have the potential to trap generations upon generations of people in
poverty [111]. Although the PROWA is a case study in successful welfare reform,
the law reformed only one of approximately seventy federal welfare programs [135].
Policymakers should consider similar reforms for these other programs. More fun-
damentally, from a policymaker’s perspective, it is always important to understand
the efficacy of policy reforms and make effective policy recommendations. Future re-
107
search should similarly examine other programs including Social Security, Medicare,
Medicaid, public housing, and the Affordable Care Act among others.
Methodologically, we have improved upon Bayesian credible interval estima-
tion by applying semiparametric density ratio estimation to truncated posterior
samples. Although this semiparametric modeling technique has had a number of
applications in frequentist models, we are the first to apply the approach to Bayesian
estimation. These methodological improvements of course need not only apply to
Bayesian models pertaining to public policy research and are worthwhile tools to be
applied in many other applied settings.
4.6 Appendix
4.6.1 An Elementary Frequentist Analysis
As an additional test of our work, we also performed a frequentist analysis
examining the impact of AFDC/TANF caseloads on discontinuances due to employ-
ment. We estimated essentially the same model we estimated earlier, but instead
made the (rather restrictive) assumption of homogeneity amongst the coefficients.
Specifically, we estimated the simple linear model:
yi,t = β0 + β1xi,t + i,t, (4.34)
where yi,t is the number of employment related discontinuances for state/territory
i at time t and xi,t′ is the number of caseloads at time t′, where t′ < t is a lagged
time. We assumed i,t ∼ N(0, 1). Our results are presented in Table 4.33.
In all five lagged comparisons, we see that the slope coefficient for TANF
programs is higher than the slope coefficient for AFDC programs, suggesting that
TANF was more successful than AFDC programs in getting people back to work.
108
Tab. 4.33: Frequentist Statistical Analysis
Coefficient Estimate Standard Error t-stat P-value Adjusted R2
AFDC Lag 1 βˆ0 0.167 0.023 7.427 < 0.001 0.524
βˆ1 0.005 0.000 28.807 < 0.001
TANF Lag 1 βˆ0 0.184 0.033 5.653 < 0.001 0.561
βˆ1 0.011 0.000 28.722 < 0.001
AFDC Lag 2 βˆ0 0.172 0.024 7.141 < 0.001 0.513
βˆ1 0.005 0.000 27.147 < 0.001
TANF Lag 2 βˆ0 0.142 0.030 4.785 < 0.001 0.636
βˆ1 0.011 0.000 32.122 < 0.001
AFDC Lag 3 βˆ0 0.177 0.026 6.878 0.144 0.495
βˆ1 0.005 0.000 25.186 < 0.001
TANF Lag 3 βˆ0 0.098 0.028 3.492 0.002 0.690
βˆ1 0.011 0.000 34.544 < 0.001
AFDC Lag 4 βˆ0 0.185 0.028 6.605 < 0.001 0.478
βˆ1 0.005 0.000 23.380 < 0.001
TANF Lag 4 βˆ0 0.086 0.030 2.859 0.004 0.684
βˆ1 0.011 0.000 32.678 < 0.001
AFDC Lag 5 βˆ0 0.197 0.031 6.407 < 0.001 0.457
βˆ1 0.005 0.000 21.314 < 0.001
TANF Lag 5 βˆ0 0.079 0.033 2.373 0.018 0.670
βˆ1 0.011 0.000 29.507 < 0.001
109
To ensure that these differences were not insignificant, we tested the null hypothesis
H0 for each lag that the slope coefficient for TANF programs, β1, was equal to the
slope coefficient for AFDC programs for that same time lag. For example, for TANF
regression results for lag 1, we tested the null hypothesis H0 that β0 = 0.005 against
the two-sided alternative Ha that β0 6= 0.005. Our results are outlined in Table 4.34
for all five lags:
Tab. 4.34: Statistical Significance of Slope Coefficients
Lag t-stat p-value
Lag 1 15.890 < 0.001
Lag 2 19.667 < 0.001
Lag 3 19.333 < 0.001
Lag 4 18.667 < 0.001
Lag 5 15.714 < 0.001
These results indicate a statistically significant difference between the slope
coefficients of the lagged TANF models compared to the coefficients corresponding
to the analogous AFDC models. These results lend further credence to the argument
that TANF programs were more successful than AFDC programs in getting people
off the welfare rolls and back to work.
110
Chapter 5: Generalized Bayesian Inferences for Counterterrorism Pol-
icy with Improved Credible Interval Estimation via Semi-
parametric Out of Sample Fusion
5.1 Introduction
5.1.1 Combating Terrorism
From the Siccari during the first century siege of Jerusalem, to the Assassins
during the 11th century in what is today’s Middle East, to the Thugs in India during
the 1400s to 1800s, to the Wall Street bombings of the early 1920s, terrorism has been
an issue endangering civilians for generations [69,126]. Today, myriads of questions
abound regarding counterterrorsim policy throughout the world including questions
pertaining to the U.S. military operations in Iraq and Afghanistan, the Troubles in
Northern Ireland, the post-Suharto years in Indonesia, and the twenty-six year long
civil war in Sri Lanka among others.
Amongst scholars in the field of international relations, terrorism is gener-
ally defined as “the deliberate use or threat of force against noncombatants by a
non-state actor in pursuit of a political goal” [15]. Understanding how to fight ter-
rorists has been an issue that policymakers throughout the world have debated and
grappled with for years. Fighting terrorists is inherently different from many of
the major conflicts of the twentieth century. During World War I and World War
II, for example, both sides had uniformed combatants representing nation-states at
war. During the Cold War, both sides understood the concept of mutually assured
destruction, and consequently did not want to be annihilated in a retaliatory strike.
Terrorists, however, are markedly different from traditional enemy combat-
ants, as they generally blend in with civilians and deliberately target innocent men,
women, and children, as well as military personnel. Many are typically brainwashed
by the belief that a better life awaits them for slaughtering their enemies [38]. As we
continue on into the twenty-first century, fighting terrorism remains an issue, and it
is important to equip policymakers with the tools to be able to do so.
A Google Scholar search of the keyword “terrorism” yields nearly 893,000
studies on the topic. Many of these studies include research statistically examining
the risk of terrorist attacks as well as looking at certain types of terrorist attacks
in particular localities across the globe [40, 71, 77, 117, 120, 155]. In 1971, Hawkes
conducted a cluster analysis of terrorism data [62]. Enders (2007) offers a compre-
hensive review of research on measuring terrorism, the efficacy of counterterroism
policies, and the causes of terrorism and its various manifestations [35]. In the 1980s,
Holden (1986, 1987) examined the “contagion effect” of American aircraft hijack-
ings [65, 66]. Enders and Sandler (1995) examine terrorist behavior from game and
choice theoretic perspectives [36]. Sandler and Arce (2003) discuss game theoretic
analyses of terrorism and their various policy implications [144]. Li and Schaub
(2005) conduct a time series analysis and examine the effect of economic global-
ization on terrorist attacks [95]. Kaplan et al (2005, 2006) look at the impact on
different “counterterror tactics,” on suicide bombings in Israel [75, 76]. Lewis et al
(2012) statistically examine the changes over time in civilian deaths in Iraq as a
result of terrorism [94]. In recent years, a number of researchers have utilized very
sophisticated statistical modeling including negative binomial distribution models
along with self-exiting hurdle models to examine the incidence as well as the prob-
112
ability of terrorist attacks [121,162,163]
Although these papers are interesting both mathematically and from a policy
perspective, these studies have typically been restricted to simply just one conflict
or one type of terrorist attack. Additionally, much of the heavily statistical research
in this area provides limited advice to policymakers about how to fortify specific
security measures to prevent various types of terrorist attacks. In this chapter, we
rigorously analyze terrorist attack data from a number of major conflicts throughout
the world. In particular, we utilize Bayesian logistic regression techniques to offer
counterterrorism strategies in four major conflicts across the globe. Kyung et al
(2011) also conducted an analysis of terrorist attack data; however, their statistical
models were not constructed in a way to offer prescriptive policy advice [90]. We not
only present a model capable of providing such advice but also improve on Bayesian
credible interval estimation techniques in the process. Our study helps shed light
on the factors that influence the success of terrorist attacks, providing policymakers
with advice on how to more strategically target their security and resources to help
them battle against this very dangerous enemy.
5.1.2 Dirichlet Process Priors and the Limitations Bayesian Parametric methods
With consistent improvements in statistical computing capabilities over the
last several decades, the use of Bayesian methods has become increasingly com-
mon in applied research. Bayesian methods are an attractive approach for modeling
real-world phenomena for a number of reasons. One reason is that the associated sta-
tistical inferences from such models are based conditionally on existing data rather
than on the distributional properties of estimators or test statistics calculated over a
long-run frequency of many imaginary unobserved samples. Additionally, Bayesian
methods provide us with “exact-sample” results, rather than being rooted in the
typical asymptotic theory that most frequentist statistical estimation methods gen-
113
erally assume. Thirdly, and perhaps most importantly, Bayesian methods enable us
to tackle problems with high-dimensional parameter spaces that would generally be
impossible to estimate from a purely frequentist perspective.
One of the most controversial aspects of Bayesian methods, however, is the
formulation of a prior distribution. Typically, prior assumptions about a model’s
parameters are made subjectively by the researcher. Often these prior distributions
belong to well-known parametric families. Normal and uniform distributions are for
instance workhorse examples of priors in the applied Bayesian statistical literature.
A common criticism with these types of parametric prior distributions, how-
ever, is that since they are subjectively chosen by the researcher their distributional
assumptions may therefore not necessarily model reality. The normal distribution,
for example, has a limited degree of flexibility as it is unimodal, does not accom-
modate skewness, and has relatively thin tails. If a normally distributed prior is a
misspecification to begin with, then misleading inferences can result. Other choices
of parametric prior distributions have similar types of limitations.
Dirichlet Process priors (hereafter referred to as DP priors) do not have these
restrictions as they allow researchers to weaken the restrictive assumptions generally
concomitant with Bayesian statistical models [4,41]. These prior distributions have
been used in a number of settings including applied economics research, health policy
research, and examining the incidence of illness among others [3, 72, 88]. DP priors
avoid the typical strict parametric assumptions regarding heterogeneity mentioned
above and instead utilize an unknown distribution G to model heterogeneity. As G
is assumed to be random, a DP Prior can be placed on this distribution. Dirichlet
Processes thus enable the researcher to place a probability distribution over a space
of probability distributions.
Mathematically, we can describe a DP prior G ∼ DP (G|G0, α) consists of two
“parameters:” G0, a parametric baseline probability measure and a concentration
114
parameter α. The baseline measure G0 can be considered to be a prior assumption
regarding the population distribution. The Dirichlet Process transforms this baseline
measure into a discrete probability distribution with the concentration parameter α
determining how close the non-parametric distribution G is to the baseline measure.
Smaller values of α indicate greater departure from the baseline measure. In the
limit, as α → ∞, G ⇒ G0 (i.e. the Dirichlet Process converges in the measure to
the baseline measure).
A commonly-used choice for the baseline measure is a normal distribution,
which the Dirichlet Process discretizes. Since discrete probability measures have
non-zero probabilities of observing identical values, they are often used for cluster
analysis. For each particular sample from an MCMC simulation, identical posterior
estimates of parameters believed to be a priori following a Dirichlet Process, are
considered to belong to the same cluster of observations. Thus, a nice aspect of
the Dirichlet Process is that it not only alleviates the strict parametric assumptions
typically associated with parametric prior distributions, but it also alleviates the
similarly restrictive assumptions associated with finite mixture models that a priori
assume a particular number of segments [22, 74].
In this chapter, we employ DP priors to weaken the generally restrictive as-
sumptions associated with commonly chosen parametric prior distributions for a
Bayesian logistic regression model. In addition, we also offer some improvements to
the credible interval estimates generated by our Bayesian estimation. We discuss
the techniques for doing so in the next section.
5.1.3 A Brief Review of Density Ratio Estimation
Density ratio estimation (referred to as DRE throughout this dissertation) is
a semiparametric modeling approach. As discussed in the previous chapter of this
dissertation, this technique has seen myriads of applications ranging from AIDS
115
research to mortality rate prediction to cluster detection among many others [43,52,
80, 81, 83]. In the univariate case, we typically assume that there are I + 1 random
samples (xi1, . . . , xini), having probability density functions gi, such that:
xij ∼ gi, i = 1, . . . , I, I + 1, j = 1, . . . , ni. (5.1)
This approach assumes that gI+1 ≡ g defines a reference density having a known ra-
tio between gi and g. In many applications, this ratio is defined to be an exponential
in terms of a vector-valued tilt function h(x):
gi(x)
g(x)
= eαi+β
′
ih(x), i = 1, . . . , I. (5.2)
By assuming a particular ratio, the statistician can estimate the parameters αi
and βi as well as the distributions of Gi and G (the CDFs of gi and g) ∀ i = 1, . . . , I
using the method of empirical likelihood. The methodology has been discussed in
detail in the previous chapter of this dissertation.
5.1.4 Using DRE to Improving on Bayesian Estimation Results
Typically, one would estimate posterior functionals using standard Markov
Chain Monte Carlo (MCMC) methods [50]. Researchers would use the resulting
MCMC samples to generate a variety of statistical estimators such as posterior
means, standard deviations, and credible interval estimates.
As discussed in Chapter 4, however, a common limitation of MCMC meth-
ods stems from computational feasibility. Most Bayesian MCMC samplers need to
be truncated in “real-time.” As a result, it is difficult to understand if we have
properly navigated the posterior distribution. Although we have some diagnostic
checks, such as autocorrelation of draws and convergence of chains [51], these tests
are merely diagnostics that may shed light on the issue, but we never completely
116
understand whether our MCMC sampler truly manifests our entire posterior distri-
bution. Therefore, point and interval estimates based on MCMC samples may not
necessarily be sufficiently accurate from which to make statistical inferences.
One can ameliorate this limitation by using the DRE Method [123–125, 160].
Although this semiparametric approach has been used in a variety of settings, this
dissertation is the first to apply the methodology in Bayesian estimation. In the
previous chapter of this dissertation, we applied the methodology to hierarchical
Bayesian linear regression. In that chapter, we illustrated, via a series of numerical
simulations, that the DRE method has the ability to provide more accurate quantiles
of distributions. In this chapter, we extend our application of this methodology to
a generalized lienar hierarchical Bayesian model. The interested reader is referred
to the previous chapter for simulation results demonstrating the efficacy of this
approach.
5.2 Problem Formulation
5.2.1 Model
Consider a data set of terrorist attacks recorded over the course of t ∈ {1, . . . , T}
discrete time occasions. We define the binary outcome variable:
yt =



1 if the terrorist attack is successful at time t
0 otherwise,
(5.3)
where pt = Prob(yt = 1) is the probability that a terrorist attack is successful. We
examined the success of terrorist attacks by estimating the following binary logistic
regression model:
117
P (Y |β) =
T∏
t=1
eyt(β0+β1Xt,1+β2Xt,2+βt,3Xt,3)
1 + eβ0+β1Xt,1+β2Xt,2+βt,3Xt,3
, (5.4)
Xt,1 =



1 if the terrorist attack at time t is a suicide attack
0 otherwise,
(5.5)
Xt,2 = (xt,2,1, . . . , xt,2,n2) is a categorically coded explanatory variable denoting the
type of terrorist attack (assassination, hijacking, bombing, etc.) Armed assaults,
defined in the appendix, are used as a benchmark variable to ensure statistical iden-
tifiability of the model. Finally, Xt,3 = (xt,3,1, . . . , xt,2,n3) is a categorically coded
explanatory variable denoting the target of the terrorist attack (airports/airlines,
business, educational institution, food or water supply, government, etc.). For the
analysis of the conflicts in Afghanistan, Iraq, Sri Lanka, and Northern Ireland after
the 1998 Good Friday Agreement, airports/airlines were the benchmark variable.
For analysis of the conflict of Northern Ireland before the 1998 Good Friday Agree-
ment, abortion related targets were the benchmark variable. β2 and βt,3 are of
course n2 and Tn3 dimensional vectors respectively.
We allow β0 ∼ N(µ0, σ20) and β1 ∼ N(µ1, σ
2
1). Additionally, we allow β2 ∼
N(µ2,Σn2×n2) and βt,3 to either follow a normally distributed prior or to follow
a DP prior. Mathematically, our two choices for varying βt,3 are either βt,3 ∼
N(µ3,ΣTn3×Tn3) or βt,3 ∼ G, where G ∼ DP (α,N(µ3,ΣTn3×Tn3)). µ0, µ1, as
well as each component of the vectors µ2 and µ3 all follow a normal distribution
with mean 0 and variance 10. σ20 and σ
2
1 are each specified to follow an inverse
gamma distribution with shape and scale parameter each equal to 10. Additionally,
for computational simplicity Σn2×n2 and ΣTn3×Tn3 are assumed to be diagonal
matrices, with each component also drawn from an inverted gamma distribution
118
with shape and scale parameter each equal to 10.
As discussed earlier, βt,3 to follow a DP prior enables us to weaken the strict
parametric assumptions associated with a normal prior distributions as well as to
cluster our analysis around the types of targets terrorists intend to attack.
5.2.2 Data
Our data sets were obtained from the START GTD Database, a database
compiled by the University of Maryland, containing detailed information on over
113,000 terrorist attacks [54]. We performed a statistical analysis on four different
conflicts - The War in Afghanistan, the War in Iraq, the Civil War in Sri Lanka, and
the Civil War in Ireland. The dependent variable in our Bayesian logistic regression
was whether or not the attack was successful, and our explanatory variables involved
whether the attack was a suicide attack, the type of attack, and the targets of the
attack. All of our explanatory variables were categorically coded. For each conflict,
a few observations could not definitively provide information regarding the details
of the attack (such as whether the attack was a suicide attack or whether the attack
was successful) and hence was excluded from our analysis.
The START database defined the success of a terrorist attack as whether the
type of attack actually took place. For example, a bombing/explosion is considered
successful if the bomb involved actually detonated. More details are contained in
the appendix to this chapter.
119
5.3 A Bayesian Analysis of Several Major Conflicts Across the
Globe
5.3.1 Estimation
We rigorously examined four recent conflicts - The War in Afghanistan, the
War in Iraq, the Civil War in Sri Lanka, and the Civil War in Northern Ireland. For
each conflict, we estimated the two Bayesian logistic regression models outlined in
Section 5.2, with one model assuming normally distributed priors for the coefficient
corresponding to the target of the terrorist attack and other assuming the less re-
strictive DP priors. We estimated both models via MCMC methods over the course
30,000 iterations, using the first half for burn in and the remaining half for statistical
inference. For the model assuming normal prior distributions, we ran our MCMC
sampler in WinBUGS [97]. For the model assuming DP priors we ran our sampler
using the R Package DPPackage [70] that used well-known algorithms for MCMC
sampling from non-conjugate priors for DP prior models [37, 98, 112]. Autocorrela-
tions of each marginal posterior sample were low, suggesting reasonable navigation
around the posterior density. As mentioned earlier, however, autocorrelations are
simply just a diagnostic check and cannot truly elicit whether the posterior density
has been adequately sampled. As a result, in addition to our standard Bayesian
analysis, we also present improvements to the Bayesian interval estimates of a few
important posterior coefficients using DRE.
5.3.2 The War in Afghanistan
On September 11, 2001, foreign terrorists struck the United States in the
most devastating attack on American soil since Pearl Harbor. A group of nineteen
terrorists hijacked three U.S. passenger planes, crashing them into the World Trade
120
Center in New York and the Pentagon in Washington, D.C. A fourth hijacked plane,
intended for the U.S. Capitol, crashed in rural Pennsylvania that same morning. A
total of over 3,000 innocent Americans died in these attacks.
Shortly thereafter, overwhelming evidence made it quite apparent that the
attacks were perpetrated by the terrorist group al Qaeda, headed at the time by
Osama bin Laden. The Taliban in Afghanistan had been harboring bin Laden,
was known for aiding and abetting al Qaeda, and was infamous for sponsoring
terrorism [79]. After the Taliban refused to hand bin Laden over to American
custody, the United States used military force to remove the Taliban from power,
began efforts toward finding bin Laden, and helped install a democratic Afghan
government that renounces terrorism [153]. Since military operations in Afghanistan
began thirteen years ago, the United States military has maintained a consistent
presence in Afghanistan, providing stability and support to help the relatively new
Afghan government.
Unfortunately, however, terrorist attacks in Afghanistan have continued over
the course of the last several years. Tables 5.1 and 5.2 present a Bayesian statis-
tical analysis of these attacks, using the modeling approach outlined above. The
data used spans from slightly after the September 11th attacks (when U.S. mili-
tary operations in Afghanistan began) through December 2011 consisting of 2887
observations.
Our results are quite informative. In Afghanistan, both models indicate that
suicide attacks, with posterior estimates around -1 in both models, were generally
unsuccessful as were assassination attempts (with posterior mean coefficient esti-
mate -3.147 in the normal model and -2.858 in the DP model), bombings (-0.541 in
the normal model and -0.634 in the DP model), attacks on infrastructure (-0.176
in the normal model and -0.418 in the DP model), and hijackings (-1.931 in the
normal model and -1.787 in the DP model.) These posterior estimates indicate the
121
Tab. 5.1: Afghanistan - Hierarchical Bayesian Model Using Normally Distributed Priors
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
Intercept 4.605 0.553 3.541 5.645
Suicide -1.281 0.236 -1.741 -0.811
Assassination -3.147 0.345 -3.841 -2.506
Bombing/Explosion -0.541 0.327 -1.191 0.081
Facility/Infrastructure -0.176 0.618 -1.321 1.110
Hijacking -1.931 1.472 -4.556 1.276
Hostage Taking - Barricade Incident 2.035 2.199 -1.543 7.024
Hostage Taking - Kidnapping 1.075 0.815 -0.352 2.855
Unknown 1.787 2.238 -1.889 6.706
Mean - Business 0.917 0.798 -0.454 2.528
Mean - Educational Institution -0.767 0.640 -2.011 0.452
Mean - Food or Water Supply 1.329 2.288 -2.595 6.176
Mean - Government (Diplomatic) -0.454 0.695 -1.757 0.958
Mean - Government (General) -0.610 0.493 -1.500 0.321
Mean - Journalists & Media 2.100 2.116 -1.270 6.734
Mean - Maritime 1.347 2.357 -2.746 6.468
Mean - Military -0.318 0.564 -1.345 0.804
Mean - NGO -0.530 0.777 -1.978 0.985
Mean - Other 0.077 1.204 -2.006 2.704
Mean - Police -0.085 0.516 -1.016 0.957
Mean - Private Citizens & Property 1.674 0.679 0.465 2.989
Mean - Religious Figures/Institutions 1.323 0.974 -0.440 3.259
Mean - Telecommunication 2.743 2.274 -0.837 7.969
Mean - Terrorists -0.381 1.089 -2.448 1.827
Mean - Tourists 0.640 2.851 -4.604 6.452
Mean - Transportation -1.085 0.618 -2.256 0.162
Mean - Unknown -2.469 0.696 -3.808 -1.111
Mean - Utilities -1.660 1.017 -3.502 0.498
Variance - Business 1.122 0.399 0.585 2.136
Variance - Educational Institution 1.148 0.406 0.601 2.148
Variance - Food or Water Supply 1.097 0.381 0.583 2.038
Variance - Government (Diplomatic) 1.110 0.394 0.587 2.068
Variance - Government (General) 1.049 0.334 0.577 1.868
Variance - Journalists & Media 1.097 0.379 0.588 2.010
Variance - Maritime 1.101 0.382 0.587 2.070
Variance - Military 1.228 0.459 0.623 2.374
Variance - NGO 1.111 0.392 0.586 2.098
Variance - Other 1.134 0.409 0.594 2.177
Variance - Police 1.032 0.322 0.577 1.814
Variance - Private Citizens & Property 1.106 0.361 0.596 2.020
Variance - Religious Figures/Institutions 1.149 0.427 0.596 2.212
Variance - Telecommunication 1.094 0.383 0.580 2.021
Variance - Terrorists 1.157 0.430 0.597 2.214
Variance - Tourists 1.109 0.388 0.581 2.054
Variance - Transportation 1.095 0.379 0.588 2.043
Variance - Unknown 1.102 0.396 0.579 2.072
Variance - Utilities 1.152 0.430 0.594 2.233
Variance - Unknown 1.112 0.402 0.581 2.109
Variance - Utilities 1.144 0.417 0.594 2.183
122
Tab. 5.2: Afghanistan - Hierarchical Bayesian Model Using DP priors
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
(Intercept) 3.965 0.348 3.289 4.664
Suicide -1.036 0.208 -1.429 -0.622
Assassination -2.858 0.299 -3.445 -2.277
Bombing/Explosion -0.634 0.314 -1.245 -0.013
Facility/Infrastructure -0.418 0.562 -1.416 0.776
Hijacking -1.787 1.225 -4.007 0.863
Hostage Taking (Barricade Incident) 1.976 2.220 -1.472 7.026
Hostage Taking (Kidnapping) 1.017 0.796 -0.363 2.820
Unknown 2.059 2.350 -1.635 7.147
µBaseline 3.853 0.635 2.569 5.086
ΣBaseline 1.007 1.052 0.214 3.286
Number of clusters 4.691 1.552 2.000 8.000
α 1.054 0.566 0.249 2.419
Business 4.863 0.655 3.539 5.942
Educational Institution 3.644 0.346 2.910 4.312
Food or Water Supply 4.084 0.826 2.694 5.781
Government (Diplomatic) 3.721 0.371 3.028 4.543
Government (General) 3.636 0.287 3.067 4.204
Journalists & Media 4.274 0.838 3.048 5.898
Maritime 4.106 0.831 2.683 5.814
Military 3.734 0.316 3.139 4.384
NGO 3.715 0.435 2.907 4.812
Other 3.948 0.666 2.923 5.556
Police 3.835 0.322 3.228 4.523
Private Citizens & Property 5.255 0.418 4.460 6.120
Religious Figures/Institutions 4.878 0.683 3.475 5.985
Telecommunication 4.424 0.843 3.174 5.965
Terrorists 3.782 0.582 2.679 5.287
Tourists 3.989 0.854 2.301 5.742
Transportation 3.519 0.421 2.526 4.222
Unknown 2.634 0.727 1.321 3.956
Utilities 3.472 0.650 1.913 4.687
success of the U.S. military in preventing terrorist attacks of this nature. Attacks
related to hostage taking, on the other hand, had positive coefficients and have
therefore generally been much more successful. As a result, it could be useful for
the U.S. military to work with Afghan security forces to determine where hostage
taking is predominately occurring and improving security around such locations.
Additionally, our model using DP priors suggests that there may be some clustering
of attacks around potential targets not necessarily observable from the data directly.
We will discuss this result in more detail later in this chapter.
5.3.3 The War in Iraq
For decades, Iraq has been a country housing three distinct ethnic and religious
groups - Sunni Muslims, Shi’ite Muslims, and Kurds. Saddam Hussein, the President
of Iraq from 1979 up until his fall in 2003, was infamous for being an egregious
violator of human rights, providing preferential treatment for many of his fellow
Sunni Muslims, while subjecting many Shi’ite Muslims and Kurds to severe and
123
often deadly persecution. In 1982, for example, Saddam detained nearly 800 men,
women, and children from the Shi’ite town of Dujail following a failed assassination
attempt. In court, prosecutors and witnesses claimed that they witnessed torture
and murder of many of these civilians. In 1988, during the final days of the Iran-
Iraq war, Saddam ordered a poison gas attack in the northern Iraqi town of Halabja,
killing thousands of innocent civilians and injuring scores of others. Many of the
survivors are still suffering from the long term effects today [64,101].
In addition to human rights violations, Saddam invaded two of his neighboring
countries, supported terrorists, led the international community to believe that he
had intentions to acquire illicit weapons, and attempted to assassinate a former U.S.
President [34, 153]. In light of these violations of international law, President Bill
Clinton in 1998 signed into law the Iraq Liberation Act, making regime change in
Iraq the explicit policy of the United States government. Four years later, Pres-
ident George W. Bush signed into law the Iraq War Resolution, allowing the use
of American ground troops to carry out the goals outlined in the Iraq Liberation
Act [5, 68].
In 2003, a military coalition led by the United States and British governments
removed Saddam from power and helped install a democratic regime that espouses
the values of freedom and human rights. Although the initial campaign in Iraq to
remove Saddam was astonishingly successful, maintaining stability in the immediate
post-Saddam Iraq was quite difficult. Terrorist attacks occurred on a regular basis,
jeopardizing the lives of coalition forces as well as civilians. Recent research has
shown that most of these suicide attacks have been targeted towards civilians [148].
Tables 5.3-5.4 contain an analysis of these attacks from 2003 through 2011, which
consisted of 7621 observations, using our Bayesian approach.
Our posterior coefficient estimates for suicide attacks illustrate that, just like
Afghanistan, suicide attacks were generally unsuccessful (with posterior mean coef-
124
Tab. 5.3: Iraq - Hierarchical Bayesian Model Using Normally Distributed Priors
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
Intercept 4.838 0.502 4.011 5.910
Suicide -0.522 0.199 -0.906 -0.130
Assassination -4.261 0.290 -4.844 -3.721
Bombing/Explosion -1.063 0.282 -1.628 -0.530
Facility/Infrastructure -0.770 0.746 -2.128 0.791
Hijacking 0.522 2.822 -4.605 6.178
Hostage Taking - Barricade Incident 2.077 2.189 -1.384 6.985
Hostage Taking - Kidnapping -0.682 0.559 -1.711 0.493
Unknown 1.265 2.475 -2.872 6.609
Mean - Business 0.677 0.573 -0.528 1.694
Mean - Educational Institution -0.536 0.606 -1.768 0.599
Mean - Food or Water Supply 2.000 2.521 -2.309 7.333
Mean - Government (Diplomatic) -0.420 0.590 -1.644 0.692
Mean - Government (General) -0.076 0.488 -1.131 0.684
Mean - Journalists & Media 1.364 0.971 -0.373 3.532
Mean - Maritime -1.709 1.514 -4.549 1.478
Mean - Military 0.112 0.510 -1.007 0.992
Mean - NGO 0.349 1.284 -1.981 2.995
Mean - Other 1.651 1.086 -0.433 3.839
Mean - Police 0.358 0.459 -0.660 1.071
Mean - Private Citizens & Property 0.998 0.493 -0.167 1.728
Mean - Religious Figures/Institutions 0.543 0.542 -0.664 1.532
Mean - Telecommunication 1.917 2.326 -1.943 7.024
Mean - Terrorists 0.027 0.594 -1.196 1.101
Mean - Tourists 1.214 2.109 -2.321 5.838
Mean - Transportation 0.280 0.614 -0.943 1.483
Mean - Unknown -1.579 0.724 -3.007 -0.143
Mean - Utilities -0.635 0.643 -1.911 0.615
Mean -Violent Political Party 3.536 1.590 0.875 7.189
Variance - Business 1.126 0.374 0.588 2.030
Variance - Educational Institution 1.176 0.419 0.601 2.201
Variance - Food or Water Supply 1.100 0.387 0.584 2.057
Variance - Government (Diplomatic) 1.226 0.459 0.618 2.349
Variance - Government (General) 1.392 0.460 0.707 2.452
Variance - Journalists & Media 1.185 0.469 0.598 2.364
Variance - Maritime 1.122 0.405 0.585 2.113
Variance - Military 1.080 0.348 0.586 1.969
Variance - NGO 1.107 0.383 0.581 2.059
Variance - Other 1.086 0.376 0.576 2.021
Variance - Police 0.831 0.229 0.479 1.323
Variance - Private Citizens & Property 1.163 0.394 0.615 2.172
Variance - Religious Figures/Institutions 1.280 0.519 0.616 2.560
Variance - Telecommunication 1.096 0.381 0.582 2.046
Variance - Terrorists 1.155 0.423 0.602 2.252
Variance - Tourists 1.101 0.384 0.584 2.055
Variance - Transportation 1.102 0.435 0.577 2.219
Variance - Unknown 1.118 0.396 0.595 2.116
Variance - Utilities 1.119 0.374 0.603 2.041
Variance -Violent Political Party 1.103 0.386 0.582 2.057
Variance - Utilities 1.135 0.411 0.587 2.150
Variance - Violent Political Party 1.107 0.393 0.586 2.064
125
Tab. 5.4: Iraq - Hierarchical Bayesian Model Using DP priors
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
(Intercept) 4.543 0.278 4.012 5.109
Suicide -0.449 0.180 -0.785 -0.087
Assassination -3.836 0.264 -4.361 -3.316
Bombing/Explosion -1.049 0.267 -1.593 -0.540
Facility/Infrastructure -1.022 0.639 -2.187 0.274
Hijacking 0.622 2.851 -4.434 6.538
Hostage Taking (Barricade Incident) 2.094 2.149 -1.361 6.976
Hostage Taking (Kidnapping) -0.645 0.517 -1.590 0.404
Unknown 1.390 2.480 -2.778 6.753
µBaseline 4.546 0.434 3.669 5.387
ΣBaseline 0.372 0.389 0.099 1.200
Number of clusters 4.685 1.566 2.000 8.000
α 1.046 0.572 0.242 2.440
Business 4.857 0.375 4.152 5.599
Educational Institution 4.251 0.368 3.473 4.961
Food or Water Supply 4.576 0.486 3.710 5.555
Government (Diplomatic) 4.257 0.347 3.548 4.939
Government (General) 4.241 0.269 3.725 4.785
Journalists & Media 4.823 0.441 3.992 5.685
Maritime 4.405 0.493 3.379 5.357
Military 4.425 0.317 3.822 5.064
NGO 4.506 0.462 3.639 5.467
Other 4.741 0.458 3.918 5.646
Police 4.633 0.281 4.097 5.199
Private Citizens & Property 5.143 0.294 4.579 5.740
Religious Figures/Institutions 4.673 0.363 3.993 5.419
Telecommunication 4.586 0.482 3.717 5.537
Terrorists 4.400 0.345 3.756 5.104
Tourists 4.563 0.482 3.689 5.528
Transportation 4.545 0.371 3.870 5.329
Unknown 4.112 0.491 2.898 4.930
Utilities 4.260 0.381 3.437 4.995
Mean -Violent Political Party 4.822 0.484 3.946 5.784
ficient estimates approximately -0.5 in both models), although not as unsuccessful
as Afghanistan. Assassination attempts (-4.261 in the normal model and -3.736 in
the DP model), bombings (-1.063 in the normal model and -1.049 in the DP model),
and attacks on Facility/Infrastructure (-0.770 in the normal model and -1.022 in the
DP model) were also generally unsuccessful. It is interesting to note, however, that
both models suggest that hijacking (0.522 in the normal model and 0.622 in the
DP model) and hostage taking via barricade incidents (2.077 in the normal model
and 2.094 in the DP model), on the other hand, were successful. Hostage taking
via kidnapping (-0.682 in the normal model and -0.645 in the DP model), however,
was not. These results suggest that Iraqi security forces that have taken respon-
sibility for the country after the departure of U.S. troops in 2011 should consider
fortifying security to prevent hijackings and hostage takings via barricade incident.
The Iraqi government could consider providing additional security personal or offer
recommendations regarding private security to do so.
126
5.3.4 The Sri Lankan Civil War
The Sri Lankan Civil War was a 26 year conflict from 1983-2009. Fought
by the Liberation Tigers of Tamil Eelam (LTTE), the war was waged against the
government of Sri Lanka, who were seeking to create a separate Tamil state within
the northeastern part of the country. The LTTE engaged in terrorist attacks rang-
ing from suicide attacks against civilians, to coordinated attacks against religious
figures and important facilities, to assassination attempts against members of the
Sri Lankan government. These attacks included the 1993 assassination of President
Ranasinghe Premadasa, the 1996 bombings of the Sri Lankan Central Bank, killing
over 90 and injuring over 1400, and the 1998 bombing of the revered Temple of
the Tooth among others [54]. Shortly after the killing of LTTE leader Velupillai
Prabhakaran in 2009, the 26 year-long Civil War ended [161]. Tables 5.5 and 5.6
contain our Bayesian analysis of this data set from over the course of the conflict
which consisted of 2924 observations.
The results from the civil war in Sri Lanka are quite similar to those in Iraq.
In particular, suicide attacks (with posterior mean coefficient estimate -0.421 in
the normal model and -0.371 in the DP model), assassination attempts (-1.487
in the normal model and -1.532 in the DP model), bombing attempts (-1.206 in
the normal model and -1.169 in the DP model), attacks on facility/infrastructure
(essentially zero in the normal model and -0.196 in the DP model), hostage taking
via kidnapping (-0.833 in the normal model and -0.877 in the DP model), unarmed
assaults (-1.206 in the normal model and -1.201 in the DP model), and terrorist
attacks of an unknown nature (-1.454 in the normal model and -1.359 in the DP
model) were largely unsuccessful. Hostage taking via barricade incident (1.405 in
the normal model and 1.848 in the DP model) and hijackings (1.822 in the normal
model and 1.848 in the DP model), on the other hand, were much more successful.
Despite the fact that the war ended in 2009, understanding these issues could be
127
Tab. 5.5: Sri Lanka - Hierarchical Bayesian Model Using Normally Distributed Priors
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
Intercept 3.842 0.570 2.867 5.018
Suicide -0.421 0.386 -1.153 0.351
Assassination -1.487 0.330 -2.148 -0.844
Bombing/Explosion -1.206 0.277 -1.752 -0.671
Facility/Infrastructure 0.006 0.628 -1.147 1.326
Hijacking 1.822 2.276 -1.933 6.908
Hostage Taking - Barricade Incident 1.405 2.448 -2.774 6.641
Hostage Taking - Kidnapping -0.833 0.659 -2.039 0.547
Unarmed Assault -1.206 1.455 -3.793 1.898
Unknown -1.454 0.431 -2.282 -0.581
Mean - Business 0.448 0.671 -0.879 1.722
Mean - Educational Institution 3.426 1.919 0.363 7.843
Mean - Food or Water Supply 2.225 2.205 -1.553 6.978
Mean - Government (Diplomatic) -1.061 0.962 -2.926 0.865
Mean - Government (General) -0.021 0.580 -1.224 1.069
Mean - Journalists & Media 0.094 0.867 -1.507 1.883
Mean - Maritime 1.100 1.169 -0.941 3.678
Mean - Military 0.580 0.575 -0.509 1.579
Mean - NGO 3.277 1.884 0.096 7.414
Mean - Other -0.701 0.920 -2.465 1.159
Mean - Police 0.416 0.602 -0.832 1.485
Mean - Private Citizens & Property 2.324 0.674 0.961 3.534
Mean - Religious Figures/Institutions 1.763 1.203 -0.340 4.471
Mean - Telecommunication -0.960 0.939 -2.785 0.895
Mean - Terrorists 3.528 1.781 0.498 7.640
Mean - Tourists 1.671 2.359 -2.406 6.703
Mean - Transportation 0.418 0.631 -0.797 1.649
Mean - Unknown -2.413 0.675 -3.776 -1.128
Mean - Utilities 0.232 0.850 -1.375 2.017
Mean - Violent Political Party 0.466 0.707 -0.876 1.857
Variance - Business 1.102 0.392 0.585 2.108
Variance - Educational Institution 1.092 0.385 0.571 2.028
Variance - Food or Water Supply 1.103 0.388 0.587 2.056
Variance - Government (Diplomatic) 1.132 0.403 0.592 2.145
Variance - Government (General) 1.141 0.430 0.591 2.157
Variance - Journalists & Media 1.117 0.389 0.585 2.099
Variance - Maritime 1.115 0.394 0.590 2.075
Variance - Military 1.149 0.452 0.591 2.313
Variance - NGO 1.096 0.385 0.582 2.050
Variance - Other 1.134 0.413 0.590 2.190
Variance - Police 1.146 0.439 0.600 2.213
Variance - Private Citizens & Property 1.083 0.359 0.574 1.997
Variance - Religious Figures/Institutions 1.094 0.382 0.577 2.070
Variance - Telecommunication 1.126 0.397 0.593 2.101
Variance - Terrorists 1.097 0.385 0.577 2.053
Variance - Tourists 1.109 0.393 0.581 2.090
Variance - Transportation 1.139 0.414 0.592 2.169
Variance - Unknown 1.180 0.437 0.602 2.271
Variance - Utilities 1.133 0.405 0.601 2.131
Variance - Violent Political Party 1.112 0.407 0.576 2.096
128
Tab. 5.6: Sri Lanka - Hierarchical Bayesian Model Using DP priors
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
(Intercept) 3.890 0.354 3.233 4.627
Suicide -0.371 0.325 -0.982 0.282
Assassination -1.532 0.300 -2.128 -0.950
Bombing/Explosion -1.169 0.263 -1.699 -0.658
Facility/Infrastructure -0.196 0.597 -1.262 1.044
Hijacking 1.848 2.151 -1.611 6.598
Hostage Taking (Barricade Incident) 1.345 2.439 -2.720 7.063
Hostage Taking (Kidnapping) -0.870 0.598 -1.948 0.414
Unarmed Assault -1.201 1.341 -3.520 1.832
Unknown -1.359 0.407 -2.121 -0.517
µBaseline 3.506 0.775 1.896 4.967
ΣBaseline 1.944 3.062 0.463 6.355
Number of clusters 4.513 1.349 3.000 8.000
α 0.999 0.529 0.263 2.306
Business 3.763 0.266 3.271 4.305
Educational Institution 4.698 0.979 3.371 6.455
Food or Water Supply 4.188 0.886 3.032 6.147
Government (Diplomatic) 3.262 0.830 1.368 4.230
Government (General) 3.669 0.284 3.053 4.191
Journalists & Media 3.742 0.369 3.027 4.473
Maritime 4.014 0.678 3.223 5.836
Military 3.783 0.241 3.333 4.284
NGO 4.572 0.973 3.336 6.374
Other 3.519 0.609 1.793 4.279
Police 3.759 0.242 3.302 4.253
Private Citizens & Property 5.452 0.465 4.598 6.410
Religious Figures/Institutions 4.456 0.875 3.357 6.159
Telecommunication 3.322 0.785 1.446 4.226
Terrorists 4.992 0.939 3.459 6.571
Tourists 4.060 0.930 2.080 6.074
Transportation 3.753 0.255 3.269 4.267
Unknown 1.604 0.454 0.716 2.499
Utilities 3.752 0.367 3.068 4.479
Violent Political Party 3.764 0.286 3.250 4.339
useful for the Sri Lankan government as post-war reconciliation processes continue.
5.3.5 The Troubles
Commonly known as “The Troubles,” the Civil War in Ireland was an ethno-
nationalist conflict over the constitutional status of Northern Ireland [102]. The
Troubles bears many similarities to the Civil War in Sri Lanka. Irish nationalists,
primarily Catholic, wanted an Independent State of Northern Ireland, while Union-
ists and loyalists, primarily Protestant, preferred Northern Ireland to remain part
of the United Kingdom. The Unionists and loyalists were represented politically
by the Ulster Volunteer Force (UVF) and the Ulster Defence Association (UDA),
while the Irish nationalists were represented by the Irish Republican Army (IRA).
Unlike many political groups, however, the UVF, UDA, and IRA were known for
engaging in violent behavior, including engaging in attacks targeting civilians. In
fact, these organizations have been deemed by the United States State Department
129
as terrorist groups alongside organizations such as al Qaeda, Hezbollah, Hamas, and
LTTE among others [158].
The Good Friday Agreement of 1998, a compromise that Northern Ireland
would remain a component of the United Kingdom until the people of northern
Ireland and the Republic of Ireland would determine otherwise, contained provisions
for the creation of institutions to civilly discuss issues between Northern Ireland
and the Republic of Ireland as well as between Britain and Ireland [114]. The
Agreement, however, was not a panacea for the country’s issues, and terrorist attacks
continued after its passage. Tables 5.7-5.10 contain our analysis of the attacks
in Northern Ireland from 1970 up through the Good Friday Agreement of 1998
(consisting of 3517 observations) and then from 1998 through the present (consisting
of 457 observations). During these time periods, only one suicide attack occurred
in our data set (on December 29, 1998 in Armagh, Northern Ireland) and was
consequently excluded from our analysis.
Our posterior estimates indicate that hostage taking via kidnapping (posterior
mean coefficient estimates 1.548 in the normal model and 1.429 in the DP model),
unarmed assaults (2.585 in the normal model and 2.619 in the DP model), and
terrorist attacks of an unknown nature (0.156 in the normal model and 2.619 in
the DP model) were the most successful types of terrorist attacks in Ireland before
the Good Friday Agreement of 1998. Although terrorist attacks continued after
the Agreement, the most successful types of attacks were via hijacking (1.146 in
the normal model and 2.869 in the DP model) and hostage taking via kidnapping
(1.090 in the normal model and 1.042 in the DP model). The success of unarmed
assaults declined after the Good Friday Agreement of 1998 as the posterior coefficient
estimates became negative (-0.865 in the normal model -1.083 in the DP model).
130
Tab. 5.7: Ireland before the 1998 Good Friday Agreement - Hierarchical Bayesian Model
Using Normally Distributed Priors
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
Intercept 2.346 0.639 1.087 3.644
Assassination -0.421 0.236 -0.900 0.028
Bombing/Explosion -0.972 0.233 -1.447 -0.524
Facility/Infrastructure -0.831 0.319 -1.450 -0.196
Hijacking -2.452 1.233 -4.837 0.056
Hostage Taking - Barricade Incident -0.283 1.268 -2.476 2.485
Hostage Taking - Kidnapping 1.548 1.113 -0.313 4.033
Unarmed Assault 2.585 2.088 -0.842 7.188
Unknown 0.156 0.861 -1.349 2.012
Mean - Airports & Airlines -1.152 1.132 -3.339 1.048
Mean - Business 0.774 0.620 -0.488 2.043
Mean - Educational Institution -0.438 0.949 -2.235 1.471
Mean - Government (Diplomatic) 1.520 2.522 -3.090 6.930
Mean - Government (General) -0.491 0.629 -1.783 0.751
Mean - Journalists & Media -0.335 1.495 -3.178 2.703
Mean - Maritime 2.158 2.310 -1.853 7.131
Mean - Military 1.067 0.647 -0.291 2.447
Mean - NGO 1.917 2.223 -2.065 6.544
Mean - Other 1.935 1.165 -0.152 4.434
Mean - Police 0.216 0.623 -1.054 1.492
Mean - Private Citizens & Property 0.898 0.616 -0.415 2.122
Mean - Religious Figures/Institutions -0.029 0.958 -1.781 1.971
Mean - Telecommunications -0.950 1.723 -4.285 2.624
Mean - Terrorists -0.279 0.669 -1.723 1.066
Mean - Tourists 1.464 2.532 -3.218 6.731
Mean - Transportation 0.543 0.756 -0.926 2.037
Mean - Unknown 1.774 0.984 0.043 3.871
Mean - Utilities -2.775 2.285 -7.589 1.418
Variance - Airports & Airlines 1.128 0.403 0.591 2.161
Variance - Business 1.195 0.428 0.619 2.306
Variance - Educational Institution 1.119 0.400 0.585 2.119
Variance - Government (Diplomatic) 1.114 0.397 0.585 2.094
Variance - Government (General) 1.340 0.536 0.654 2.729
Variance - Journalists & Media 1.131 0.408 0.593 2.164
Variance - Maritime 1.103 0.381 0.585 2.041
Variance - Military 1.162 0.444 0.586 2.326
Variance - NGO 1.106 0.391 0.579 2.082
Variance - Other 1.116 0.394 0.589 2.113
Variance - Police 1.210 0.428 0.631 2.324
Variance - Private Citizens & Property 1.001 0.335 0.547 1.866
Variance - Religious Figures/Institutions 1.124 0.406 0.590 2.144
Variance - Telecommunications 1.120 0.405 0.591 2.121
Variance - Terrorists 1.070 0.365 0.576 1.987
Variance - Tourists 1.104 0.386 0.584 2.084
Variance - Transportation 1.134 0.410 0.593 2.148
Variance - Unknown 1.113 0.389 0.582 2.103
Variance - Utilities 1.111 0.394 0.587 2.092
131
Tab. 5.8: Ireland before the 1998 Good Friday Agreement - Hierarchical Bayesian Model
Using DP priors
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
(Intercept) 2.336 0.240 1.854 2.802
Assassination -0.393 0.195 -0.768 -0.001
Bombing/Explosion -0.897 0.191 -1.259 -0.530
Facility/Infrastructure -0.847 0.253 -1.314 -0.360
Hijacking -2.282 1.075 -4.222 -0.064
Hostage Taking (Barricade Incident) -0.012 1.174 -2.052 2.263
Hostage Taking (Kidnapping) 1.429 0.997 -0.308 3.603
Unarmed Assault 2.619 2.193 -0.859 7.858
Unknown 0.222 0.817 -1.192 1.955
µBaseline 2.231 0.396 1.453 2.998
ΣBaseline 0.391 0.350 0.113 1.187
Number of clusters 4.453 1.441 2.000 8.000
α 1.006 0.546 0.234 2.320
Airports & Airlines 1.977 0.491 1.162 3.007
Business 2.783 0.203 2.359 3.167
Educational Institution 2.021 0.472 1.262 3.002
Government (Diplomatic) 2.379 0.536 1.365 3.174
Government (General) 1.703 0.250 1.228 2.213
Journalists & Media 2.236 0.542 1.299 3.118
Maritime 2.419 0.526 1.399 3.179
Military 2.831 0.185 2.476 3.195
NGO 2.420 0.530 1.392 3.183
Other 2.689 0.375 1.766 3.240
Police 2.152 0.201 1.757 2.545
Private Citizens & Property 2.821 0.184 2.467 3.182
Religious Figures/Institutions 2.191 0.510 1.330 3.086
Telecommunications 2.194 0.547 1.260 3.099
Terrorists 1.810 0.279 1.291 2.362
Tourists 2.366 0.543 1.359 3.168
Transportation 2.555 0.391 1.728 3.145
Unknown 2.782 0.291 2.088 3.273
Utilities 2.144 0.554 1.214 3.090
Tab. 5.9: Ireland after the 1998 Good Friday Agreement - Hierarchical Bayesian Model
Using Normally Distributed Priors
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
Intercept 2.819 1.058 0.910 4.946
Assassination 0.462 1.114 -1.734 2.530
Bombing Explosion -2.260 1.049 -4.355 -0.379
Facility/Infrastructure 3.759 2.077 -0.005 8.227
Hijacking 1.146 2.613 -3.549 6.608
Hostage Taking - Kidnapping 1.090 1.203 -1.318 3.368
Unarmed Assault -0.865 1.648 -3.965 2.529
Mean - Business -0.226 0.591 -1.361 0.962
Mean - Educational Institution 0.223 0.482 -0.716 1.162
Mean - Government (General) 2.191 1.224 0.158 5.149
Mean - Journalists & Media -2.353 2.281 -7.232 1.800
Mean - Military -0.680 1.573 -3.670 2.601
Mean - NGO 1.068 2.654 -3.878 6.628
Mean - Other -1.390 1.829 -4.945 2.298
Mean - Police -0.889 1.135 -3.037 1.455
Mean - Private Citizens & Property -0.209 0.388 -1.013 0.512
Mean - Religious Figures/Institutions 0.668 1.322 -1.743 3.554
Mean - Terrorists 3.080 1.941 -0.166 7.442
Mean - Tourists -1.516 1.224 -3.942 0.885
Mean - Transportation -1.288 0.908 -3.060 0.476
Mean - Unknown -3.902 1.071 -6.157 -1.986
Mean - Utilities -2.514 2.379 -7.595 1.825
Mean - Violent Political Party 2.808 2.037 -0.659 7.306
Variance - Business 1.148 0.402 0.602 2.147
Variance - Educational Institution 1.085 0.367 0.580 1.998
Variance - Government (General) 1.109 0.392 0.581 2.082
Variance - Journalists & Media 1.112 0.398 0.581 2.101
Variance - Military 1.106 0.391 0.585 2.078
Variance - NGO 1.108 0.390 0.584 2.077
Variance - Other 1.114 0.398 0.590 2.084
Variance - Police 1.143 0.406 0.594 2.149
Variance - Private Citizens & Property 1.136 0.410 0.599 2.169
Variance - Religious Figures/Institutions 1.097 0.380 0.585 2.037
Variance - Terrorists 1.100 0.387 0.584 2.077
Variance - Tourists 1.110 0.392 0.590 2.067
Variance - Transportation 1.111 0.391 0.589 2.102
Variance - Unknown 1.087 0.374 0.577 2.021
Variance - Utilities 1.109 0.392 0.585 2.073
Variance - Violent Political Party 1.106 0.387 0.588 2.069
132
Tab. 5.10: Ireland after the 1998 Good Friday Agreement - Hierarchical Bayesian Model
Using DP priors
Post. Post. 95% Credible Interval 95% Credible Interval
Mean Std Dev Lower Limit Upper Limit
(Intercept) 2.417 0.474 1.514 3.372
Assassination -0.210 1.268 -2.336 2.821
Bombing/Explosion -2.288 0.414 -3.159 -1.542
Facility/Infrastructure 0.475 0.742 -0.863 2.117
Hijacking 2.869 1.964 -0.496 7.233
Hostage Taking (Kidnapping) 1.042 2.514 -3.330 6.289
Unarmed Assault -1.083 1.368 -3.460 2.115
µBaseline 1.796 0.885 -0.109 3.435
ΣBaseline 1.212 1.564 0.181 4.792
Number of clusters 3.272 1.291 2.000 6.000
α 0.798 0.491 0.154 2.031
Business 2.737 0.422 1.995 3.633
Educational Institution 2.453 0.776 0.423 3.618
Government (General) 2.685 0.426 1.891 3.547
Journalists & Media 2.543 0.806 0.377 3.857
Military 2.909 0.646 2.024 4.669
NGO 2.284 0.945 -0.108 3.580
Other 2.470 0.724 0.546 3.568
Police 2.738 0.417 2.000 3.621
Private Citizens & Property 2.685 0.396 1.981 3.514
Religious Figures/Institutions 2.723 0.521 1.780 3.807
Terrorists 2.839 0.640 1.955 4.504
Tourists 2.238 0.911 0.007 3.507
Transportation 2.278 0.808 0.357 3.468
Unknown 0.529 0.901 -1.337 2.226
Utilities 2.155 1.077 -0.507 3.568
Violent Political Party 2.798 0.624 1.872 4.262
5.3.6 The Use of Dirichlet Process Priors
We applied either a normal prior or a DP Prior to the coefficients regarding
the targets of terrorist attacks. To compare the DP model for each conflict with its
normal counterpart, we estimated pseudo-Bayes factors (PsBF) [47,48]. The PsBF
is based on a model’s cross-validation predictive density. As standard Bayes factors
are often not computationally feasible to compute, the PsBF is considered a useful
surrogate. If we let y be our observed data, yt be the tth terrorist attack at time t
across t = 1, . . . , T observations, and y(t) be the data of attacks with observation t
deleted, the cross validative predictive density pi(yt|y(t)) is:
pi(yt|y(t)) =
∫
pi(yt|β,y(t))pi(β, |y(t))dβ
≈
[
1
R
R∑
r=1
1
f(yt,β
(r))
]−1
, (5.6)
where f is the likelihood function and β(r) is the vector of parameter values ob-
133
tained during the rth MCMC iteration [115]. The relation in (5.6) is based on a
truncated series approximation of the harmonic mean of the logistic regression func-
tion evaluated at each posterior draw, averaged across all R post burn-in MCMC
iterations [115]. As a result, the PsBF comparing the normal model (M=1) to the
DP model (M=2) can be written as follows:
PsBF =
T∏
t=1
pi(yt|y(t),M = 2)
pi(yt|y(t),M = 1)
. (5.7)
We can take advantage of the additive properties of logarithms and utilize
(5.6) to estimate the logarithms of the numerator as well as the denominator in
(5.11). These quantities, approximations to the log-marginal data likelihoods for
each model, enable us to estimate PsBFs for each conflict examined. Table 5.11
contains our results.
Tab. 5.11: PsBF Computation comparing DP Model to Normal Model
Normal Model DP Model Pseudo Bayes Factor
Afghanistan -518.066 -510.610 1730.213
Iraq -1183.940 -1183.310 1.878
Sri Lanka -563.147 -556.940 496.310
N. Ireland Before 1998 GF Agreement -1278.738 -1272.590 467.781
N. Ireland After 1998 GF Agreement -230.315 -231.140 0.438
PsBF estimates greater than one indicate stronger support for the DP Model,
whereas estimates less than one provide support for the normal model. Furthermore,
the larger the value, the stronger the indication of support [47, 48].1 Therefore,
these results indicate substantial support for the DP Model for understanding the
determinants of successful attacks in Afghanistan, Sri Lanka, and Northern Ireland
before the 1998 Good Friday Agreement. The PsBF for Iraq is 1.878, indicated some
support for the DP Model over the normal model for that conflict. Thus, for four
1 Pseudo-Bayes Factors have the potential to legitimately take on extremely large values (as
large as exp(50)) that would indicate overwhelming evidence in favor of one model over another,
as illustrated in Ansari and Mela’s (2003) hierarchical Bayesian probit analysis of online customer
clickstream behavior [3].
134
of the five conflicts examined, the DP model effectively captures clustering around
potential targets for these conflicts compared to the normal model. The normal
model, on the other hand, is considerably more adequate for modeling the conflict
in Northern Ireland after the 1998 Good Friday Agreement. The adequacy of the
normal model compared to the DP Model in this instance may be due the lack of
clustering of attacks around targets in Northern Ireland in recent years.
Table 5.12 contains our average cluster sizes for each conflict, averaged over
our MCMC iterations:
Tab. 5.12: Average Cluster Size - DP prior Model
Average Cluster Size Standard Deviation
Afghanistan 4.691 1.552
Iraq 4.685 1.566
Sri Lanka 4.513 1.349
Northern Ireland (before 1998 GF Agreement) 4.453 1.441
Northern Ireland (after 1998 GF Agreement) 3.272 1.291
These results suggest an interesting phenomenon illustrated by our DP prior
model. In particular, our models cluster around a small number of targets. This clus-
tering manifests the terrorists’ strategy, improving the model’s explanatory power for
three of the conflicts described above. Future research could look into determining
the exact composition of these clusters and provide military advice accordingly. Re-
cent research has performed similar cluster analysis of other real-world phenomena,
including the topics discussed by Barack Obama, John McCain, and Mitt Romney
in the last two presidential elections [113].
5.3.7 Improving Bayesian Credible Interval Estimates
In the previous chapter of this dissertation, we discussed utilizing the semi-
parametric DRE method to improve the credible interval estimation of a hierarchical
linear model regarding welfare reform. In this section, we extend this technique to
refine the credible interval estimates of the hierarchical generalized linear model
135
Fig. 5.1: Histogram of Posterior Sample for Suicide Attack Coefficient in Afghanistan,
Assuming a Normally Distributed Prior
of this chapter. Specifically, we took posterior samples of several coefficients that
warranted a more rigorous examination of their distributional properties. Upon
looking at the histograms of these posterior samples, they all seemed to be reason-
ably “normal-like” in nature, appearing to be unimodal and reasonably symmetric
with limited skewness. Several of these histograms are in Figures 5.1-5.10
As a result of these marginal posterior samples’ distributional behavior, we
used the approach in the previous chapter to estimate the 95% credible interval,
again assuming a tilt function of the form h(x) = (x, x2)′. As the simulations of
the previous chapter illustrated, applying the DRE method to posterior samples
can provide more accurate quantile estimation. Our results are presented in Tables
5.13-5.17:
As we saw with our rigorous examination of welfare reform, the semiparametric
DRE method enables us to better understand our posterior interval estimates re-
136
Fig. 5.2: Histogram of Posterior Sample for Suicide Attack Coefficient in Afghanistan,
Assuming a DP prior
Tab. 5.13: Bayesian Credible Interval Refinements for Suicide Attack Posterior Coefficient,
Afghanistan
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
Normal Prior -1.741 -0.811 -1.748 -0.817
DP Prior -1.429 -0.622 -1.439 -0.626
Tab. 5.14: Bayesian Credible Interval Refinements for Bombing/Explosion Posterior Co-
efficient, Iraq
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
Normal Prior -1.628 -0.530 -1.622 -0.521
DP Prior -1.593 -0.540 -1.571 -0.528
Tab. 5.15: Bayesian Credible Interval Refinements for Facility/Infrastructure Posterior
Coefficient, Sri Lanka
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
Normal Prior -1.147 1.326 -1.206 1.275
DP Prior -1.262 1.044 -1.339 1.010
137
Fig. 5.3: Histogram of Posterior Sample for Bombing/Explosion Attack Coefficient in Iraq,
Assuming a Normally Distributed Prior
Tab. 5.16: Bayesian Credible Interval Refinements for Hijacking Posterior Coefficient,
Northern Ireland before 1998 Good Friday Agreement
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
Normal Prior -4.837 0.056 -4.878 0.028
DP Prior -4.222 -0.064 -4.342 -0.132
Tab. 5.17: Bayesian Credible Interval Refinements for Unarmed Assault Posterior Coeffi-
cient, Northern Ireland bafterefore 1998 Good Friday Agreement
95% Credible Interval 95% Credible Interval Refined 95% Credible Refined 95% Credible
Lower Limit Upper Limit Interval Lower Limit Interval Upper Limit
Normal Prior -3.965 2.529 -4.072 2.431
DP Prior -3.460 2.115 -3.672 1.737
138
Fig. 5.4: Histogram of Posterior Sample for Bombing/Explosion Attack Coefficient in Iraq,
Assuming a DP prior
garding terrorist attacks. This information is useful for policymakers to understand
upper and lower limits of response coefficients so they can make more accurate
statistical inferences regarding fortification of appropriate security measures. For
example, in Northern Ireland after the 1998 Good Friday Agreement, although the
success of unarmed assaults remains relatively low, the upper limit of the pertinent
marginal posterior distribution still remains positive. This result suggests that this
positive upper interval estimate was unlikely due to truncation of the MCMC sam-
pler. We can use these results to advise associated policymakers that regardless of
the negative Bayesian point estimator generated pertaining to unarmed assaults,
they should not neglect security issues regarding attacks of this nature.
139
Fig. 5.5: Histogram of Posterior Sample for Facility/Infrastructure Coefficient in Sri
Lanka, Assuming a Normally Distributed Prior
5.4 Conclusions and Future Research
Our results provide policymakers with a model offering useful counterterrorism
strategies shedding light on which types of attacks are sufficiently defended against
and which others warrant further fortification. As a result, policymakers can use our
model in conjunction with other statistical models aimed at understanding terror-
ist networks as well as theoretical research in political science examining the goals,
causes, and onset of terrorism [2, 15, 89, 120, 143, 166]. It is interesting to note that
across the conflicts studied here, suicide attacks are generally unsuccessful, despite
the tremendous attention they garner from the mainstream media. This result is
consistent with the findings of Kyung et al (2011)’s study of the Middle East and
Northern Africa [90]. Other considerably more coordinated non-suicide attacks,
however, such as hostage taking, are often much more successful. This phenomenon
140
Fig. 5.6: Histogram of Posterior Sample for Facility/Infrastructure Coefficient in Sri
Lanka, Assuming a DP prior
can be explained by the fact that security forces in these countries may be successful
at deterring terrorist attacks, although further measures need to be taken to deter
other types of attacks.
There are many extensions of this research that we hope will aid policymak-
ers. For example, policy researchers could potentially use this type of model on a
considerably more local level as different regions within a country will almost surely
require different security measures. The Iraq troop surge of 2007, for example, was
intended to quell violence in Baghdad as well as the Al-Anbar province [12]. A more
micro analysis, perhaps at a provincial level, can help policymakers understand what
exactly such security measures may be necessary. Another potential avenue for fu-
ture research could be to estimate a similar Bayesian statistical model presented
here and instead examine the impact of different types of weapons on the success
of terrorist attacks. This information is available in the rich START database and
141
Fig. 5.7: Histogram of Posterior Sample for Hijacking Coefficient in Northern Ireland (be-
fore 1998 GF Agreement), Assuming a Normally Distributed Prior
could provide valuable information to policymakers.
Methodologically, there is also a significant amount of future research that
we hope that this study will encourage. In particular, four of the eight models in
this study applied standard DP priors to weaken the typically restrictive parametric
assumptions associated with Bayesian statistical models. An interesting avenue of
future research is to apply distance-dependent DP priors, where the mechanism for
clustering is guided based on distance between terrorist attacks [8]. Bayesian infer-
ences from models of this nature could provide useful military advice for combating
terrorist attacks in the future.
There is also of course no a priori reason to restrict this type of modeling
to counterterrorism policy as there are myriads of other applications of Bayesian
models ranging in applied economics, business, and professional sports among oth-
ers [1,3,72]. In this study, we also utilized the semiparametric DRE method to pro-
142
Fig. 5.8: Histogram of Posterior Sample for Hijacking Coefficient in Northern Ireland (be-
fore 1998 GF Agreement), Assuming a DP prior
vide more accurate estimation of several Bayesian credible intervals for our posterior
coefficients. An interesting avenue of future research could be to use this approach to
provide detailed estimates for posterior densities corresponding to other parametric
and non-parametric models. Additionally, in this study, we only applied the DRE
method to marginal posterior distributions. Another potentially interesting avenue
for future research could be to apply this approach to develop Bayesian credible
regions for multivariate posterior densities. Finally, we applied this approach to
posterior samples, based on significantly larger samples generated by MCMC meth-
ods. For high-dimensional models involving large data sets, adequately sampling
a posterior density can be quite time consuming and computationally burdensome.
Generalized direct sampling is a faster alternative to MCMC that generates inde-
pendent samples and can more quickly navigate the posterior [10]. Although these
independent samples are quite small compared to MCMC samples, associated sta-
143
Fig. 5.9: Histogram of Posterior Sample for Unarmed Assault Coefficient in Northern Ire-
land (after 1998 GF Agreement), Assuming a Normally Distributed Prior
tistical inferences based on such samples can be significantly improved upon by
utilizing the semiparametric DRE method.
5.5 Appendix
5.5.1 Variable Definition
Below are descriptions of the variables used in this chapter, as defined in the
GTD Codebook [55]. Our dependent variable was whether the terrorist attack was
successful.
Dependent Variable Used in Model - Successful attack
144
Fig. 5.10: Histogram of Posterior Sample for Unarmed Assault Coefficient in Northern
Ireland (after 1998 GF Agreement), Assuming a DP prior
Success - “Success of a terrorist strike is defined according to the tangible effects of
the attack. Success is not judged in terms of the larger goals of the perpetra-
tors. For example, a bomb that exploded in a building would be counted as
a success even if it did not succeed in bringing the building down or inducing
government repression.” Please see descriptions under each of the individual
explanatory variables below about how this definition is applied to each type
of terrorist attack.
Explanatory Variable Used in Model - Suicide Attack
Suicide - “This variable is coded ‘Yes’ in those cases where there is evidence that
the perpetrator did not intend to escape from the attack alive.”
145
Explanatory Variable Used in Model - Types of Terrorist Attacks
Assassination - “An act whose primary objective is to kill one or more specific,
prominent individuals. Usually carried out on persons of some note, such as
high-ranking military officers, government officials, celebrities, etc. Not to
include attacks on non-specific members of a targeted group. The killing of a
police officer would be an armed assault unless there is reason to believe the
attackers singled out a particularly prominent officer for assassination ... In
order for an assassination to be successful, the target of the assassination must
be killed. For example, even if an attack kills numerous people but not the
target, it is an unsuccessful assassination.”
Armed Assault - “An attack whose primary objective is to cause physical harm or
death directly to human beings by use of a firearm, incendiary, or sharp in-
strument (knife, etc.). Also included under this attack type would be CBRN
(chemical, biological, radiological, nuclear) weapons. Not to include acts of
a purely personal or criminal nature, or acts in which people are incidentally
harmed in pursuit of another primary objective. Not to include attacks involv-
ing the use of fists, rocks, sticks, or other handheld (less-than-lethal) weapons
... An armed assault is determined to be successful if the assault takes place
and if a target is hit. Unsuccessful armed assaults are those in which the
perpetrators attack and do not hit the target. An armed assault is also un-
successful if the perpetrators are apprehended on their way to commit the
assault. To make this determination, however, there must be information to
indicate that an actual assault was imminent.”
Bombing/Explosion - “An attack where the primary effects are caused by an en-
ergetically unstable material undergoing rapid decomposition and releasing a
pressure wave that causes physical damage to the surrounding environment.
146
Can include either high or low explosives (including a dirty bomb) but does
not include a nuclear explosive device that releases energy from fission and/or
fusion, or an incendiary device where decomposition takes place at a much
slower rate ... A bombing is successful if the bomb or explosive device det-
onates. Bombings are considered unsuccessful if they do not detonate. The
success or failure of the bombing is not based on whether it hit the intended
target.”
Hijacking - “An act whose primary objective is to take control of a vehicle such as an
aircraft, boat, bus, etc. for the purpose of diverting it to an unprogrammed
destination, force the release of prisoners, or some other political objective.
Obtaining payment of a ransom should not the sole purpose of a Hijacking,
but can be one element of the incident so long as additional objectives have
also been stated. Hijackings are distinct from Hostage Taking because the
target is a vehicle, regardless of whether there are people/passengers in the
vehicle ... A hijacking is successful if the hijackers assume control of the vehicle
at any point, whereas a hijacking is unsuccessful if the hijackers fail to assume
control of the vehicle. The success or failure of the hijacking is not based on
whether the vehicle reached the intended destination of the hijackers.”
Hostage Taking (Barricade Incident) - “An act whose primary objective is to take
control of hostages for the purpose of achieving a political objective through
concessions or through disruption of normal operations. Such attacks are
distinguished from kidnapping since the incident occurs and usually plays out
at the target location with little or no intention to hold the hostages for an
extended period in a separate clandestine location ... A barricade incident is
successful if the hostage takers assume control of the individuals at any point,
whereas a barricade incident is unsuccessful if the hostage takers fail to assume
147
control of the individuals.”
Hostage Taking (Kidnapping) - “An act whose primary objective is to take con-
trol of hostages for the purpose of achieving a political objective through
concessions or through disruption of normal operations. Kidnappings are dis-
tinguished from Barricade Incidents (above) in that they involve moving and
holding the hostages in another location ... A kidnapping is successful if the
kidnappers assume control of the individuals at any point, whereas a kidnap-
ping is unsuccessful if the kidnappers fail to assume control of the individuals.”
Facility/Infrastructure - “An act, excluding the use of an explosive, whose primary
objective is to cause damage to a non-human target, such as a building, mon-
ument, train, pipeline, etc. Such attacks include arson and various forms of
sabotage (e.g., sabotaging a train track is a Facility/Infrastructure, even if
passengers are killed). Facility/Infrastructures can include acts which aim to
harm an installation, yet also cause harm to people incidentally (e.g. an arson
attack primarily aimed at damaging a building, but causes injuries or fatalities)
... A facility attack is determined to be successful if the facility is damaged.
If the facility has not been damaged, then the attack is unsuccessful.”
Unarmed Assault - “An attack whose primary objective is to cause physical harm
or death directly to human beings by any means other than explosive, firearm,
incendiary, or sharp instrument (knife, etc.) ... An unarmed assault is deter-
mined to be successful there is a victim that who has been injured. Unarmed
assaults that are unsuccessful are those in which the perpetrators do not in-
jure anyone ... An unarmed assault is also unsuccessful if the perpetrators are
apprehended when on their way to commit the assault. To make this deter-
mination, however, there must be information to indicate that an assault was
imminent.”
148
Explanatory Variable Used in Model - Targets of Terrorist Attacks
Business - “Businesses are defined as individuals or organizations engaged in com-
mercial or mercantile activity as a means of livelihood. Any attack on a busi-
ness or private citizens patronizing a business such as a restaurant, gas station,
music store, bar, caf, etc. This includes attacks carried out against corporate
offices or employees of firms like mining companies, or oil corporations. Fur-
thermore, includes attacks conducted on business people or corporate officers.
Included in this value as well are hospitals and chambers of commerce and
cooperatives. Does not include attacks carried out in public or quasi-public
areas such as business district or commercial area, (these attacks are captured
under Private Citizens and Property, see below.)”
Government (General) - “Any attack on a government building; government mem-
ber, former members, including members of political parties in official capac-
ities, their convoys, or events sponsored by political parties; political move-
ments; or a government sponsored institution where the attack is expressly
carried out to harm the government. This value includes attacks on judges,
public attorneys (e.g., prosecutors), courts and court systems, politicians, roy-
alty, head of state, government employees (unless police or military), election-
related attacks, intelligence agencies and spies, or family members of govern-
ment officials when the relationship is relevant to the motive of the attack.”
Police - “This value includes attacks on members of the police force or police in-
stallations; this includes police boxes, patrols headquarters, academies, cars,
checkpoints, etc. Includes attacks against jails or prison facilities, or jail or
prison staff or guards.”
Military - “Includes attacks against army units, patrols, barracks, and convoys,
149
jeeps, etc. Also includes attacks on recruiting sites, and soldiers engaged in
internal policing functions such as at checkpoints and in anti-narcotics activi-
ties. Excludes attacks against non-state militias and guerrillas, these types of
attacks are coded as Terrorist/Non-state Militias see below.”
Abortion Related - “Attacks on abortion clinics, employees, patrons, or security
personnel stationed at clinics.”
Airports and Airlines - “An attack that was carried out either against an airplane or
against an airport. Attacks against airline employees while on board are also
included in this value. Includes attacks conducted against airport business
offices and executives. Attacks where airplanes were used to carry out the
attack (such as three of the four 9/11 attacks) are not included.”
Government (Diplomatic) - “Attacks carried out against foreign missions, includ-
ing embassies, consulates, etc. This value includes cultural centers that have
diplomatic functions, and attacks against diplomatic staff and their families
(when the relationship is relevant to the motive of the attack) and property.
The United Nations is a diplomatic target.”
Educational Institution - “Attacks against schools, teachers, or guards protecting
school sites. Includes attacks against university professors, teaching staff and
school buses. Moreover, includes attacks against religious schools in this value.
As noted below in the Private Citizens and Property value, the GTD has sev-
eral attacks against students. If attacks involving students are not expressly
against a school, university or other educational institution or are carried out
in an educational setting, they are coded as private citizens and property. Ex-
cludes attacks against military schools (attacks on military schools are coded
as Military, see below).”
150
Food or Water Supply - “Attacks on food or water supplies or reserves are included
in this value. This generally includes attacks aimed at the infrastructure re-
lated to food and water for human consumption.”
Journalists and Media - “Includes, attacks on reporters, news assistants, photogra-
phers, publishers, as well as attacks on media headquarters and offices. Attacks
on transmission facilities such as antennae or transmission towers, or broadcast
infrastructure are coded as Telecommunications, see below.”
Maritime - “Implies civilian maritime. Includes attacks against fishing ships, oil
tankers, ferries, yachts, etc. (Attacks on fishermen are coded as Private Citi-
zens and Property, see below).”
NGO - “Includes attacks on offices and employees of non-governmental organiza-
tions (NGOs). NGOs here include large multinational non-governmental or-
ganizations such as the Red Cross and Doctors without Borders. Does not
include labor unions, social clubs, student groups, and other non-NGO (such
cases are coded as Other, see immediately below).”
Other - “This value includes acts of terrorism committed against targets which do
not fit into other categories.”
Private Citizens and Property - “This value includes attacks on individuals, the
public in general or attacks in public areas including markets, commercial
streets, busy intersections and pedestrian malls. Also includes ambiguous cases
where the target/victim was a named individual, or where the target/victim of
an attack could be identified by name, age, occupation, gender or nationality.
This value also includes ceremonial events, such as weddings and funerals.
The GTD contains a number of attacks against students. If these attacks are
not expressly against a school, university or other educational institution or
151
are not carried out in an educational setting, these attacks are coded using this
value. Also, includes incidents involving political supporters as private citizens
and property, provided that these supporters are not part of a government-
sponsored event. Finally, this value includes police informers. Does not include
attacks causing civilian casualties in businesses such as restaurants, cafes or
movie theaters (these categories are coded as Business see above).”
Religious Figures and Institutions - “This value includes attacks on religious lead-
ers, (Imams, priests, bishops, etc.), religious institutions (mosques, churches),
religious places or objects (shrines, relics, etc.). This value also includes at-
tacks on organizations that are affiliated with religious entities that are not
NGOs, businesses or schools. Attacks on religious pilgrims are considered Pri-
vate Citizens and Property; attacks on missionaries are considered religious
figures.”
Telecommunication - “This includes attacks on facilities and infrastructure for the
transmission of information. More specifically this value includes things like
cell phone towers, telephone booths, television transmitters, radio, and mi-
crowave towers.”
Terrorists/Non-State Militias - “Terrorists or members of identified terrorist groups
within the GTD are included in this value. Membership is broadly defined and
includes informants for terrorist groups, but excludes former or surrendered
terrorists. This value also includes cases involving the targeting of militias and
guerillas.”
Tourists - “This value includes the targeting of tour buses, tourists, or ‘tours.’
Tourists are persons who travel primarily for the purposes of leisure or amuse-
ment. Government tourist offices are included in this value. The attack must
152
clearly target tourists, not just an assault on a business or transportation
system used by tourists. Travel agencies are coded as business targets.
Transportation (Other than aviation) - “Attacks on public transportation systems
are included in this value. This can include efforts to assault public buses,
minibuses, trains, metro/subways, highways (if the highway itself is the target
of the attack), bridges, roads, etc. The GTD contains a number of attacks on
generic terms such as cars or vehicles. These attacks are assumed to be against
Private Citizens and Property unless shown to be against public transportation
systems. In this regard, buses are assumed to be public transportation unless
otherwise noted.”
Unknown - “The target type cannot be determined from the available information.”
Utilities - “This value pertains to facilities for the transmission or generation of
energy. For example, power lines, oil pipelines, electrical transformers, high
tension lines, gas and electric substations, are all included in this value. This
value also includes lampposts or street lights. Attacks on officers, employees
or facilities of utility companies excluding the type of faculties above are coded
as business.”
Violent Political Parties - “This value pertains to entities that are both political
parties (and thus, coded as ’government’ in this coding scheme) and terrorists.
It is operationally defined as groups that engage in electoral politics and appear
as Perpetrators in the GTD.”
153
Chapter 6: Conclusions and Future Research - Where do we go from
here?
This dissertation has offered a number of improvements to Bayesian statistical
methodologies alongside a number of contributions to public policy research. There
are still, however, many other important questions future research should look at. In
particular, the tilt functions used in the semiparametric density ratio models called
upon in this thesis as well as in other research are chosen a priori by the researcher.
Future research could look into a systematic manner, perhaps via variational calcu-
lus, of determining the optimal tilt functions. There are also many other statistical
methodologies that warrant future research including improvements to other non-
MCMC based Bayesian estimation techniques, such as variational Bayesian meth-
ods, particle filtering, and generalized direct sampling [6,10,87]. Fast alternatives to
MCMC methods are of utmost importance to researchers as the need for analyzing
large data sets has grown tremendously in recent years in a number of applied fields.
Additionally, there are myriads of other questions beyond the ones looked at
here that could benefit from rigorous analysis, including questions regarding edu-
cation policy, energy policy, immigration policy, health policy, and macroeconomic
modeling. Methodologies similar to those utilized in this dissertation, as well as
many of the techniques mentioned above, could be particularly useful in looking at
these questions and others.
We emphasize, however, that the ideas presented in this dissertation are not
applicable just to public policy research. Regardless of the application, it is always
important to be able to properly model real-world phenomena. We hope that this
dissertation provides a number of useful tools for applied statisticians to use.
155
Bibliography
[1] Albright, S. C. A statistical analysis of hitting streaks in baseball. Journal
of the American Statistical Association 88, 424 (1993), 1175–1183.
[2] Allanach, J., Tu, H., Singh, S., Willett, P., and Pattipati, K.
Detecting, tracking, and counteracting terrorist networks via hidden Markov
models. In Aerospace Conference, 2004. Proceedings. 2004 IEEE (2004), vol. 5,
IEEE.
[3] Ansari, A., and Mela, C. F. E-customization. Journal of Marketing
Research 40, 2 (2003), 131–145.
[4] Antoniak, C. E., et al. Mixtures of Dirichlet processes with applications
to Bayesian nonparametric problems. The Annals of Statistics 2, 6 (1974),
1152–1174.
[5] Authorization for Use of Military Force Against Iraq Resolu-
tion of 2002. Public Law 107-243, 116 Stat. 1498, 2002.
[6] Beal, M. J. Variational algorithms for approximate Bayesian inference.
Ph.D. Dissertation, University of London, 2003.
[7] Bender, R., and Grouven, U. Using binary logistic regression models for
ordinal data with non-proportional odds. Journal of Clinical Epidemiology 51,
10 (1998), 809–816.
[8] Blei, D. M., and Frazier, P. I. Distance dependent Chinese restaurant
processes. The Journal of Machine Learning Research 12 (2011), 2461–2488.
[9] Bradlow, E. T., Hardie, B. G., and Fader, P. S. Bayesian inference
for the negative binomial distribution via polynomial expansions. Journal of
Computational and Graphical Statistics 11, 1 (2002).
[10] Braun, M., and Damien, P. Generalized direct sampling for hierarchical
Bayesian models. arXiv preprint arXiv:1108.2245 (2011).
[11] Bucklin, R. E., and Gupta, S. Brand choice, purchase incidence, and seg-
mentation: An integrated modeling approach. Journal of Marketing Research
29, 2 (1992).
[12] Bush, G. W. President’s address to the nation. http:
//georgewbush-whitehouse.archives.gov/news/releases/2007/01/
20070110-7.html, January 2007. [Online; accessed April 17, 2014].
[13] Capretta, J., and Dayaratna, K. D. Compelling evidence makes the
case for a market-driven health care system. Heritage Foundation Back-
grounder, 2867 (2013).
[14] Castles, F. G., and Leibfried, S. The Oxford Handbook of the Welfare
State. Oxford Handbooks Online, 2010.
[15] Chenoweth, E. Terrorism and democracy. Annual Review of Political Sci-
ence 16 (2013), 355–378.
[16] Clinton, B. My Life. Random House LLC, 2005.
[17] Congressional Budget Office. The effects of tort reform: Evidence from
the states. A CBO Paper (June 2004).
[18] Cornyn, J., and Meese, E. Health care and medical malpractice reform:
The necessity of reform in the current debate. Heritage Foundation Lecture
1142 .
[19] Crain, N., Crain, M., McQuillan, L. J., and Abramyan, H. Tort law
tally: How state tort reforms affect tort losses and tort insurance premiums.
Pacific Research Institute, San Francisco, CA (2009).
[20] Dayaratna, K. D. Studies show: Medicaid patients have worse access and
outcomes than the privately insured. Heritage Foundation Backgrounder, 2740
(2012).
[21] Dayaratna, K. D. Competitive markets in health care: The next revolution.
Heritage Foundation Backgrounder, 2833 (2013).
[22] Dayaratna, K. D., and Kannan, P. K. A mathematical reformulation
of the reference price. Marketing Letters 23, 3 (2012), 839–849.
[23] Dayaratna, K. D., and Miller, S. J. First order approximations of
the Pythagorean won-loss formula for predicting MLB teams’ winning per-
centages. By The Numbers: The Newsletter of the SABR Statistical Analysis
Committee 22, 1 (2012), 15–19.
[24] Dayaratna, K. D., and Miller, S. J. The Pythagorean won-loss formula
and hockey: A statistical justification for using the classic baseball formula as
an evaluative tool in hockey. The Hockey Research Journal: A Publication of
the Society for International Hockey Research (2013), 193–209.
[25] Department of Health and Human Services, Administration for
Children and Families, Office of Family Assistance. Temporary
Assistance for Needy Families Program: Eighth annual report to Congress.
157
[26] Department of Health and Human Services, Administration for
Children and Families, Office of Family Assistance. Temporary
Assistance for Needy Families Program: Fifth annual report to Congress.
[27] Department of Health and Human Services, Administration for
Children and Families, Office of Family Assistance. Temporary
Assistance for Needy Families Program: Fourth annual report to Congress.
[28] Department of Health and Human Services, Administration for
Children and Families, Office of Family Assistance. Temporary
Assistance for Needy Families Program: Ninth annual report to Congress.
[29] Department of Health and Human Services, Administration for
Children and Families, Office of Family Assistance. Temporary
Assistance for Needy Families Program: Second annual report to Congress.
[30] Department of Health and Human Services, Administration for
Children and Families, Office of Family Assistance. Temporary
Assistance for Needy Families Program: Seventh annual report to Congress.
[31] Department of Health and Human Services, Administration for
Children and Families, Office of Family Assistance. Temporary
Assistance for Needy Families Program: Sixth annual report to Congress.
[32] Department of Health and Human Services, Administration for
Children and Families, Office of Family Assistance. Temporary
Assistance for Needy Families Program: Third annual report to Congress.
[33] Department of Health and Human Services, Administration
for Children and Families, Office of Family Assistance. Acf
archives. http://archive.acf.hhs.gov/programs/ofa/data-reports/
afdc/rpt3800/3800index.htm, 1994-1997. [Online; accessed April 17, 2014].
[34] Drehle, D. V., and Smith, J. R. U.S. strikes Iraq for plot to kill Bush.
The Washington Post (June 1993).
[35] Enders, W. Terrorism: An empirical analysis. Handbook of Defense Eco-
nomics 2 (2007), 815–866.
[36] Enders, W., and Sandler, T. Terrorism: Theory and applications. Hand-
book of Defense Economics 1 (1995), 213–249.
[37] Escobar, M. D., and West, M. Bayesian density estimation and inference
using mixtures. Journal of the American Statistical Association 90, 430 (1995),
577–588.
[38] Esposito, J. L. Unholy War: Terror in the Name of Islam. Oxford Univer-
sity Press, 2003.
158
[39] Everson, P. J., and Bradlow, E. T. Bayesian inference for the beta-
binomial distribution via polynomial expansions. Journal of Computational
and Graphical Statistics 11, 1 (2002).
[40] Ezell, B. C., Bennett, S. P., Von Winterfeldt, D., Sokolowski,
J., and Collins, A. J. Probabilistic risk analysis and terrorism risk. Risk
Analysis 30, 4 (2010), 575–589.
[41] Ferguson, T. S. A Bayesian analysis of some nonparametric problems. The
Annals of Statistics (1973), 209–230.
[42] Fienberg, S. E. Bayesian models and methods in public policy and govern-
ment settings. Statistical Science 26, 2 (2011), 212–226.
[43] Fokianos, K. Merging information for semiparametric density estimation.
Journal of the Royal Statistical Society: Series B (Statistical Methodology) 66,
4 (2004), 941–958.
[44] Fokianos, K., Kedem, B., Qin, J., and Short, D. A. A semiparametric
approach to the one-way layout. Technometrics 43, 1 (2001).
[45] Fokianos, K., and Qin, J. A note on Monte Carlo maximization by the
density ratio model. Journal of Statistical Theory and Practice 2, 3 (2008),
355–367.
[46] Garner, B. A., and Black, H. C. Black’s Law Dictionary. St. Paul, MN:
Thomson/West, 2004.
[47] Geisser, S., and Eddy, W. F. A predictive approach to model selection.
Journal of the American Statistical Association 74, 365 (1979), 153–160.
[48] Gelfand, A. E. Model determination using sampling-based methods. In
Markov chain Monte Carlo in practice. Springer, 1996, pp. 145–161.
[49] Gelfand, A. E., Hills, S. E., Racine-Poon, A., and Smith, A. F.
Illustration of Bayesian inference in normal data models using Gibbs sampling.
Journal of the American Statistical Association 85, 412 (1990), 972–985.
[50] Gelfand, A. E., and Smith, A. F. Sampling-based approaches to calcu-
lating marginal densities. Journal of the American Statistical Association 85,
410 (1990), 398–409.
[51] Gelman, A., and Rubin, D. B. Inference from iterative simulation using
multiple sequences. Statistical Science (1992), 457–472.
[52] Gilbert, P. B., Lele, S. R., and Vardi, Y. Maximum likelihood estima-
tion in semiparametric selection bias models with application to AIDS vaccine
trials. Biometrika 86, 1 (1999), 27–43.
159
[53] Gingrich, N. Real Change: From the World that Fails to the World that
Works. Regnery Publishing, 2008.
[54] Global Terrorism Database [Data file]. National Consortium for the
Study of Terrorism and Responses to Terrorism (START). 2011.
[55] Global Terrorism Database [Data file]. START GTD Database Code-
book: Inclusion Criteria and Variables, October 2012.
[56] Greenspan, A. The Map and the Territory: Risk, Human Nature, and the
Future of Forecasting. Penguin, 2013.
[57] Grimmer, J. An introduction to Bayesian inference via variational approxi-
mations. Political Analysis 19, 1 (2011), 32–47.
[58] Guadagni, P. M., and Little, J. D. A logit model of brand choice
calibrated on scanner data. Marketing Science 2, 3 (1983), 203–238.
[59] Guadagni, P. M., and Little, J. D. When and what to buy: A nested
logit model of coffee purchase. Journal of Forecasting 17, 3-4 (1998), 303–326.
[60] Gupta, S., and Chintagunta, P. K. On using demographic variables to
determine segment membership in logit mixture models. Journal of Marketing
Research (1994), 128–136.
[61] Haskins, R., and Greenberg, M. Welfare Reform: Success or Failure?
Policy and Practice 64, 1 (2006), 10.
[62] Hawkes, A. G. Spectra of some self-exciting and mutually exciting point
processes. Biometrika 58, 1 (1971), 83–90.
[63] Hederman, R. S., and Rector, R. The Role of Parental Work in Child
Poverty. Heritage Foundation Center for Data Analysis Report 03, 01 (2003).
[64] Hiltermann, J. R. A Poisonous Affair: America, Iraq, and the Gassing of
Halabja. Cambridge University Press, 2007.
[65] Holden, R. T. The contagiousness of aircraft hijacking. American Journal
of Sociology (1986), 874–904.
[66] Holden, R. T. Time series analysis of a contagious process. Journal of the
American Statistical Association 82, 400 (1987), 1019–1026.
[67] Hymowitz, K., Carroll, J., Wilcox, W., and Kaye, K. Knot
Yet: The Benefits and Costs of Delayed Marriage in America. http:
//twentysomethingmarriage.org/, 2013. [Online; accessed April 17, 2014].
[68] Iraq Liberation Act of 1998. Public Law 105–338, 112 Stat. 3178, 1998.
160
[69] Jackson, O. A. The impact of the 9/11 terrorist attacks on the US economy.
Journal of 9/11 Studies 20 (2008), 1–27.
[70] Jara, A., Hanson, T. E., Quintana, F. A., Mu¨ller, P., and Rosner,
G. L. DPpackage: Bayesian semi and nonparametric modeling in R. Journal
of Statistical Software 40, 5 (2011), 1.
[71] Jha, M. K. Dynamic Bayesian network for predicting the likelihood of a
terrorist attack at critical transportation infrastructure facilities. Journal of
Infrastructure Systems 15, 1 (2009), 31–39.
[72] Jochmann, M., and Leon-Gonzalez, R. Estimating the demand for
health care with panel data: A semiparametric Bayesian approach. Health
Economics 13, 10 (2004), 1003–1014.
[73] Jones, A. Should tort reform be part of health care reform? A “liberal”
thinks so. The Wall Street Journal (May 2010).
[74] Kamakura, W. A., and Russell, G. J. A probabilistic choice model for
market segmentation and elasticity structure. Journal of Marketing Research
(1989), 379–390.
[75] Kaplan, E. H., Mintz, A., and Mishal, S. Tactical prevention of suicide
bombings in Israel. Interfaces 36, 6 (2006), 553–561.
[76] Kaplan, E. H., Mintz, A., Mishal, S., and Samban, C. What happened
to suicide bombings in Israel? Insights from a terror stock model. Studies in
Conflict & Terrorism 28, 3 (2005), 225–235.
[77] Kaplan, S. Applying the general theory of quantitative risk assessment
(QRA) to terrorism risk. Risk-Based Decision Making (2002), 77–81.
[78] Katzoff, M., Zhou, W., Khan, D., Lu, G., and Kedem, B. Out of
sample fusion in risk prediction. Journal of Statistical Theory and Practice
Forthcoming (2013).
[79] Kean, T. The 9/11 commission report: Final report of the national commis-
sion on terrorist attacks upon the United States. Government Printing Office,
2011.
[80] Kedem, B., Kim, E.-y., Voulgaraki, A., and Graubard, B. I. Two-
dimensional semiparametric density ratio modeling of testicular germ cell data.
Statistics in Medicine 28, 16 (2009), 2147–2159.
[81] Kedem, B., Lu, G., Wei, R., and Williams, P. D. Forecasting mortality
rates via density ratio modeling. Canadian Journal of Statistics 36, 2 (2008),
193–206.
161
[82] Kedem, B., Pan, L., and Zhou, W. Out of sample fusion: A Monte Carlo
method. Proceedings of the 2014 IEEE Conference on Computational Science
and Computational Intelligence (2014), 364–367.
[83] Kedem, B., and Wen, S. Semi-parametric cluster detection. Journal of
Statistical Theory and Practice 1, 1 (2007), 49–72.
[84] Kennedy, D. M., Cohen, L., and Bailey, T. A. The American Pageant.
Houghton Mifflin Harcourt, 2005.
[85] Kessler, D., and McClellan, M. Do doctors practice defensive medicine?
The Quarterly Journal of Economics 111, 2 (1996), 353–390.
[86] Kessler, D. P., Sage, W. M., and Becker, D. J. Impact of malpractice
reforms on the supply of physician services. Journal of the American Medical
Association 293, 21 (2005), 2618–2625.
[87] Khan, Z., Balch, T., and Dellaert, F. MCMC-based particle filtering
for tracking a variable number of interacting targets. Pattern Analysis and
Machine Intelligence, IEEE Transactions on 27, 11 (2005), 1805–1819.
[88] Kottas, A., Duan, J. A., and Gelfand, A. E. Modeling disease inci-
dence data with spatial and spatio temporal Dirichlet process mixtures. Bio-
metrical Journal 50, 1 (2008), 29–42.
[89] Kydd, A. H., and Walter, B. F. The strategies of terrorism. International
Security 31, 1 (2006), 49–80.
[90] Kyung, M., Gill, J., and Casella, G. New findings from terrorism data:
Dirichlet process random-effects models for latent groups. Journal of the Royal
Statistical Society: Series C (Applied Statistics) 60, 5 (2011), 701–721.
[91] Lazar, N. A. Bayesian empirical likelihood. Biometrika 90, 2 (2003), 319–
326.
[92] Lenk, P. J., and DeSarbo, W. S. Bayesian inference for finite mixtures
of generalized linear models with random effects. Psychometrika 65, 1 (2000),
93–119.
[93] Levy, R. Do’s and dont’s of tort reform. Cato Institute Commentary (May
2005).
[94] Lewis, E., Mohler, G., Brantingham, P. J., and Bertozzi, A. L.
Self-exciting point process models of civilian deaths in Iraq. Security Journal
25, 3 (2012), 244–264.
[95] Li, Q., and Schaub, D. Economic globalization and transnational terrorism:
A pooled time-series analysis. Journal of Conflict Resolution 48, 2 (2004),
230–258.
162
[96] Lu, G. Asymptotic theory for multiple-sample semiparametric density ratio
models and its application to mortality forecasting. Ph.D. Dissertation, Uni-
versity of Maryland, 2007.
[97] Lunn, D. J., Thomas, A., Best, N., and Spiegelhalter, D. WinBUGS
- A Bayesian modelling framework: concepts, structure, and extensibility.
Statistics and Computing 10, 4 (2000), 325–337.
[98] MacEachern, S. N., and Mu¨ller, P. Estimating mixture of Dirichlet
process models. Journal of Computational and Graphical Statistics 7, 2 (1998),
223–238.
[99] McFadden, D. Conditional logit analysis of qualitative choice behavior.
Frontiers in Econometrics (1973), 105–142.
[100] McFadden, D., and Train, K. Mixed MNL models for discrete response.
Journal of Applied Econometrics 15, 5 (2000), 447–470.
[101] McGeough, P. Witness won’t let Saddam intimidate him. The Sydney
Morning Herald (December 2005).
[102] McKittrick, D., and McVea, D. Making Sense of the Troubles: The
Story of the Conflict in Northern Ireland. New Amsterdam Books, 2002.
[103] McQuillan, L. J. Jackpot Justice: The True Cost of America’s Tort System.
Pacific Research Institute, 2007.
[104] McQuillan, L. J., and Abramyan, H. U.S. tort liability index: 2010
report. Pacific Research Institute, San Francisco, CA (2010).
[105] McShane, B., Adrian, M., Bradlow, E. T., and Fader, P. S. Count
models based on Weibull interarrival times. Journal of Business & Economic
Statistics 26, 3 (2008).
[106] Mengersen, K. L., Pudlo, P., and Robert, C. P. Bayesian computa-
tion via empirical likelihood. Proceedings of the National Academy of Sciences
110, 4 (2013), 1321–1326.
[107] Miller, S. J. A derivation of the Pythagorean Won-Loss Formula in baseball.
Chance 20, 1 (2007), 40–48.
[108] Miller, S. J., Bradlow, E. T., and Dayaratna, K. Closed-form
Bayesian inferences for the logit model via polynomial expansions. Quan-
titative Marketing and Economics 4, 2 (2006), 173–206.
[109] Miller, S. J., and Takloo-Bighash, R. An Invitation to Modern Number
Theory, vol. 13. Princeton University Press, 2006.
[110] Morris, C. N. Parametric empirical Bayes inference: theory and applica-
tions. Journal of the American Statistical Association 78, 381 (1983), 47–55.
163
[111] Murray, C. Losing Ground: American Social Policy, 1950-1980. Basic
books, 2008.
[112] Neal, R. M. Markov chain sampling methods for Dirichlet process mixture
models. Journal of Computational and Graphical Statistics 9, 2 (2000), 249–
265.
[113] Nguyen, V.-A., Boyd-Graber, J., Resnik, P., Cai, D. A., Midberry,
J. E., and Wang, Y. Modeling topic control to detect influence in conver-
sations using nonparametric topic models. Machine Learning (2013), 1–41.
[114] Northern Ireland Peace Agreement (The Good Friday Agree-
ment). http://peacemaker.un.org/sites/peacemaker.un.org/files/
IE%20GB_980410_Northern%20Ireland%20Agreement.pdf, April 1998. [On-
line; accessed April 19, 2014].
[115] Ntzoufras, I. Bayesian Modeling using WinBUGS, vol. 698. John Wiley &
Sons, 2011.
[116] Owen, A. B. Empirical likelihood ratio confidence intervals for a single
functional. Biometrika 75, 2 (1988), 237–249.
[117] Pate´-Cornell, E. Fusion of intelligence information: A Bayesian approach.
Risk Analysis 22, 3 (2002), 445–454.
[118] Personal Responsibility and Work Opportunity Reconciliation
Act of 1996. Public Law 104–193, 110 Stat. 2105, 1996.
[119] Phue, J.-N., Kedem, B., Jaluria, P., and Shiloach, J. Evaluating
microarrays using a semiparametric approach: Application to the central car-
bon metabolism of Escherichia coli BL21 and JM109. Genomics 89, 2 (2007),
300–305.
[120] Popp, R. L., and Yen, J. Emergent Information Technologies and Enabling
Policies for Counter-terrorism, vol. 6. John Wiley & Sons, 2006.
[121] Porter, M. D., White, G., et al. Self-exciting hurdle models for terrorist
activity. The Annals of Applied Statistics 6, 1 (2012), 106–124.
[122] Prentice, R. L., and Pyke, R. Logistic disease incidence models and
case-control studies. Biometrika 66, 3 (1979), 403–411.
[123] Qin, J., and Lawless, J. Estimating equations, empirical likelihood and
constraints on parameters. Canadian Journal of Statistics 23, 2 (1995), 145–
159.
[124] Qin, J., and Zhang, B. A goodness-of-fit test for logistic regression models
based on case-control data. Biometrika 84, 3 (1997), 609–618.
164
[125] Qin, J., and Zhang, B. Density estimation under a two-sample semipara-
metric model. Nonparametric Statistics 17, 6 (2005), 665–683.
[126] Rapoport, D. C. Fear and trembling: Terrorism in three religious traditions.
The American Political Science Review (1984), 658–677.
[127] Reagan, R. A Time for Choosing: The Speeches of Ronald Reagan, 1961-
1982. Gateway Books, 1983.
[128] Rector, R. Undermining true welfare reform. Heritage Foundation Com-
mentary 14, 3, 283–298.
[129] Rector, R. Welfare reform, dependency reduction, and labor market entry.
Journal of Labor Research 14, 3 (1993), 283–297.
[130] Rector, R. Marriage: America’s greatest weapon against child poverty.
Heritage Foundation Backgrounder, 2465 (2010).
[131] Rector, R., Bradley, K., and Sheffield, R. Obama to spent $10.3
trillion on welfare. Heritage Foundation Special Report SR, 67 (2009).
[132] Rector, R., and Fagan, P. The continuing good news about welfare
reform. Heritage Foundation Backgrounder, 1620 (2003).
[133] Rector, R., and Johnson, K. Roles of couples’ relationship skills and
fathers’ employment in encouraging marriage. Heritage Foundation Center
for Data Analysis Report 04, 14 (2004).
[134] Rector, R., Johnson, K., and Noyes, L. R. Teenage sexual abstinence
and academic achievement. Heritage Foundation Center for Data Analysis
Report 03, 04 (2003).
[135] Rector, R., and Marshall, J. The unfinished work of welfare reform.
National Affairs (2013).
[136] Rector, R., and Sheffield, R. Understanding poverty in the United
States: Surprising facts about America’s poor. Heritage Foundation Back-
grounder, 2607 (2011).
[137] Rector, R., and Sheffield, R. Ending work for welfare: An overview.
Heritage Foundation Issue Brief, 3712 (2012).
[138] Revelt, D., and Train, K. Mixed logit with repeated choices: households’
choices of appliance efficiency level. Review of Economics and Statistics 80, 4
(1998), 647–657.
[139] Rivlin, A. Curing health care: the next president should complete, not
abandon, Obama’s reform. Brookings Institution Campaign 2012 Policy Brief
(May 2012).
165
[140] Rossi, P. E., and Allenby, G. M. Bayesian Statistics and Marketing.
Marketing Science 22, 3 (2003), 304–328.
[141] Rossi, P. E., Allenby, G. M., and McCulloch, R. E. Bayesian Statis-
tics and Marketing. J. Wiley & Sons, 2005.
[142] Rubin, P. H., and Shepherd, J. M. Tort reform and accidental deaths.
Journal of Law and Economics 50, 2 (2007), 221–238.
[143] Sanchez-Cuenca, I., and De la Calle, L. Domestic terrorism: The
hidden side of political violence. Annual Review of Political Science 12 (2009),
31–49.
[144] Sandler, T., et al. Terrorism & game theory. Simulation & Gaming 34,
3 (2003), 319–337.
[145] Sawhill, I. V., and Haskins, R. Work and Marriage: The Way to End
Poverty and Welfare. Welfare Reform & Beyond Initiative, Brookings Insti-
tution, 2003.
[146] Schennach, S. M. Bayesian exponentially tilted empirical likelihood.
Biometrika 92, 1 (2005), 31–46.
[147] Schmittlein, D. C., Morrison, D. G., and Colombo, R. Counting
your customers: Who-are they and what will they do next? Management
Science 33, 1 (1987), 1–24.
[148] Seifert, K. R., and McCauley, C. Suicide bombers in Iraq, 2003–2010:
disaggregating targets can reveal insurgent motives and priorities. Terrorism
and Political Violence Forthcoming .
[149] Shiryaev, A. N. Probability. Springer-Verlag, New York, 1996.
[150] Smith, K. R., and Zick, C. D. The incidence of poverty among the recently
widowed: Mediating factors in the life course. Journal of Marriage and the
Family (1986), 619–630.
[151] Studdert, D. M., Mello, M. M., Sage, W. M., DesRoches, C. M.,
Peugh, J., Zapert, K., and Brennan, T. A. Defensive medicine among
high-risk specialist physicians in a volatile malpractice environment. Journal
of the American Medical Association 293, 21 (2005), 2609–2617.
[152] Tanner, M., and DeHaven, T. TANF and federal welfare. Downsizing
the Federal Government (September 2010).
[153] Tenet, G., and Harlow, B. At the Center of the Storm: My Years at the
CIA. HarperPress, 2007.
[154] Tierney, L. Markov chains for exploring posterior distributions. The Annals
of Statistics (1994), 1701–1728.
166
[155] Tranchita, C., Hadjsaid, N., and Torres, A. Ranking contingency
resulting from terrorism by utilization of Bayesian networks. In Electrotechni-
cal Conference, 2006. MELECON 2006. IEEE Mediterranean (2006), IEEE,
pp. 964–967.
[156] Unicon Research Corporation [producer and distributor of CPS
Utilities]. Current Population Surveys, March 1962-2009 [machine-readable
data files]/conducted by the Bureau of the Census for the Bureau of Labor
Statistics. Washington: Bureau of the Census [producer and distributor]. Los
Angeles, CA., 2009.
[157] US Department of Health and Human Services and others. Quar-
terly Public Assistance Statistics. Washington DC: HHS, Administration on
Children and Families (1982-1993).
[158] US Department of State Office of the Coordinator for Coun-
terterrorism. Country Reports on Terrorism 2004. http://www.state.
gov/documents/organization/45313.pdf, April 2005. [Online; accessed
April 17, 2014].
[159] Voulgaraki, A. Semiparametric Regression and Mortality Rate Prediction.
Ph.D. Dissertation, University of Maryland, 2011.
[160] Voulgaraki, A., Kedem, B., Graubard, B. I., et al. Semiparametric
regression in testicular germ cell data. The Annals of Applied Statistics 6, 3
(2012), 1185–1208.
[161] Weaver, M., and Chamberlain, G. Sri Lanka declares end to war with
Tamil Tigers. The Guardian 19 (2009).
[162] White, G., and Porter, M. D. GPU accelerated MCMC for modeling
terrorist activity. Computational Statistics & Data Analysis 71 (2014), 643–
651.
[163] White, G., Porter, M. D., and Mazerolle, L. Terrorism Risk, Re-
silience and Volatility: A Comparison of Terrorism Patterns in Three South-
east Asian Countries. Journal of Quantitative Criminology 29, 2 (2013), 295–
320.
[164] Wisner, D. H. A stepwise logistic regression analysis of factors affecting
morbidity and mortality after thoracic trauma: effect of epidural analgesia.
Journal of Trauma-Injury, Infection, and Critical Care 30, 7 (1990), 799–805.
[165] Yang, Y., He, X., et al. Bayesian empirical likelihood for quantile regres-
sion. The Annals of Statistics 40, 2 (2012), 1102–1131.
[166] Young, J. K., and Findley, M. G. Promise and pitfalls of terrorism
research. International Studies Review 13, 3 (2011), 411–431.
167
[167] Zhang, J., and Kai, F. Y. What’s the relative risk?: A method of correcting
the odds ratio in cohort studies of common outcomes. Journal of the American
Medical Association 280, 19 (1998), 1690–1691.
[168] Zhou, W. Out of Sample Fusion. Ph.D. Dissertation, University of Maryland,
2012.
168