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Bayesian Methods
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Table of Contents

BACKGROUND AND INTRODUCTION
Introduction
Motivation and Justification
Why Are We Uncertain about Probability?
Bayes' Law
Conditional Inference with Bayes' Law
Historical Comments
The Scientific Process in Our Social Sciences
Introducing Markov Chain Monte Carlo Techniques
Exercises

SPECIFYING BAYESIAN MODELS
Purpose
Likelihood Theory and Estimation
The Basic Bayesian Framework
Bayesian "Learning"
Comments on Prior Distributions
Bayesian versus Non-Bayesian Approaches
Exercises
Computational Addendum: R for Basic Analysis

THE NORMAL AND STUDENT'S-T MODELS
Why Be Normal?
The Normal Model with Variance Known
The Normal Model with Mean Known
The Normal Model with Both Mean and Variance Unknown
Multivariate Normal Model, and S Both Unknown
Simulated Effects of Differing Priors
Some Normal Comments
The Student's t Model
Normal Mixture Models
Exercises
Computational Addendum: Normal Examples

THE BAYESIAN LINEAR MODEL
The Basic Regression Model
Posterior Predictive Distribution for the Data
The Bayesian Linear Regression Model with Heteroscedasticity
Exercises
Computational Addendum

THE BAYESIAN PRIOR
A Prior Discussion of Priors
A Plethora of Priors
Conjugate Prior Forms
Uninformative Prior Distributions
Informative Prior Distributions
Hybrid Prior Forms
Nonparametric Priors
Bayesian Shrinkage
Exercises

ASSESSING MODEL QUALITY
Motivation
Basic Sensitivity Analysis
Robustness Evaluation
Comparing Data to the Posterior Predictive Distribution
Simple Bayesian Model Averaging
Concluding Comments on Model Quality
Exercises
Computational Addendum

BAYESIAN HYPOTHESIS TESTING AND THE BAYES' FACTOR
Motivation
Bayesian Inference and Hypothesis Testing
The Bayes' Factor as Evidence
The Bayesian Information Criterion (BIC)
The Deviance Information Criterion (DIC)
Comparing Posteriors with the Kullback-Leibler Distance
Laplace Approximation of Bayesian Posterior Densities
Exercises

Bayesian Decision Theory
Introducing Decision Theory
Basic Definitions
Regression-Style Models with Decision Theory
James-Stein Estimation
Empirical Bayes
Exercises

Monte Carlo and Related Iterative Methods
Background
Basic Monte Carlo Integration
Rejection Sampling
Classical Numerical Integration
Gaussian Quadrature
Importance Sampling/Sampling Importance Resampling
Mode Finding and the EM Algorithm
Survey of Random Number Generation
Concluding Remarks
Exercises
Computational Addendum: R Code for Importance Sampling

BASICS OF MARKOV CHAIN MONTE CARLO
Who Is Markov and What Is He Doing with Chains?
General Properties of Markov Chains
The Gibbs Sampler
The Metropolis-Hastings Algorithm
The Hit-and-Run Algorithm
The Data Augmentation Algorithm
Historical Comments
Exercises
Computational Addendum: Simple R Graphing Routines for
MCMC

Implementing Bayesian Models with Markov Chain Monte Carlo
Introduction to Bayesian Software Solutions
It's Only a Name: BUGS
Model Specification with BUGS
Differences between WinBUGS and JAGS Code
Technical Background about the Algorithm
Epilogue
Exercises

BAYESIAN HIERARCHICAL MODELS
Introduction to Multilevel Models
Standard Multilevel Linear Models
A Poisson-Gamma Hierarchical Model
The General Role of Priors and Hyperpriors
Exchangeability
Empirical Bayes
Exercises
Computational Addendum: Instructions for Running JAGS, Trade Data Model

SOME MARKOV CHAIN MONTE CARLO THEORY
Motivation
Measure and Probability Preliminaries
Specific Markov Chain Properties
Defining and Reaching Convergence
Rates of Convergence
Implementation Concerns
Exercises

UTILITARIAN MARKOV CHAIN MONTE CARLO
Practical Considerations and Admonitions
Assessing Convergence of Markov Chains
Mixing and Acceleration
Producing the Marginal Likelihood Integral from Metropolis-
Hastings Output
Rao-Blackwellizing for Improved Variance Estimation
Exercises
Computational Addendum: R Code for the Death Penalty Support Model and BUGS Code for the Military Personnel Model

Markov Chain Monte Carlo Extensions
Simulated Annealing
Reversible Jump Algorithms
Perfect Sampling
Exercises

APPENDIX A: GENERALIZED LINEAR MODEL REVIEW
Terms
The Generalized Linear Model
Numerical Maximum Likelihood
Quasi-Likelihood
Exercises
R for Generalized Linear Models

APPENDIX B: COMMON PROBABILITY DISTRIBUTIONS

REFERENCES

AUTHOR INDEX
SUBJECT INDEX

About the Author

Jeff Gill is a professor in the Department of Political Science, the Division of Biostatistics, and the Department of Surgery (Public Health Sciences) at Washington University. He is the author of several books and has published numerous research articles. His research applies Bayesian modeling and data analysis to questions in general social science quantitative methodology, political behavior and institutions, and medical/health data analysis using computationally intensive tools. He received his B.A. from UCLA, MBA from Georgetown University, Ph.D. from American University, and Post-Doctorate from Harvard University.

Reviews

Praise for the Third Edition:
Bayesian Methods covers a broad yet essential scope of topics necessary for one to understand and conduct applied Bayesian analysis. The numerous social science examples should resonate with the target audience, and the availability of the code and data in an R package, BaM, further enhances the appeal of the book.
-The American Statistician, 2016

Praise for the Second Edition:
The book will be very suitable for students of social science ... The reference list is carefully compiled; it will be very useful for a well-motivated reader. Altogether it is a very readable book, based on solid scholarship and written with conviction, gusto, and a sense of fun.
-International Statistical Review (2009), 77, 2

The second edition of Bayesian Methods: A Social and Behavioral Sciences Approach is a major update from the original version. ... The result is a general audience text suitable for a first course in Bayesian statistics at the upper undergraduate level for highly quantitative students or at the graduate level for students in a wider variety of fields. ... Of the texts I have tried so far in [my] class, Gill's book has definitely worked the best for me. ... this book fills an important market segment for classes where the canonical Bayesian texts are a bit too advanced. The emphasis is on using Bayesian methods in practice, with topics introduced via higher-level discussions followed by implementation and theory. ...
-Herbert K.H. Lee, University of California, Santa Cruz, The American Statistician, November 2008

Praise for the First Edition:
This book is a brilliant and importantly very accessible introduction to the concept and application of Bayesian approaches to data analysis. The clear strength of the book is in making the concept practical and accessible, without necessarily dumbing it down. ... The coverage is also remarkable.
-S.V. Subramanian, Harvard School of Public Health

One of the contributions of Bayesian Methods: A Social and Behavioral Sciences Approach is to reintroduce Bayesian inference and computing to a general social sciences audience. This is an important contribution-one that will make demand for this book high ... Jeff Gill has gone some way toward reinventing the graduate-level methodology textbook ... Gill's treatment of the practicalities of convergence is a real service ... new users of the technique will appreciate this material. ... the inclusion of material on hierarchical modeling at first seems unconventional; its use in political science, while increasing, has been limited. However, Bayesian inference and MCMC methods are well suited to these types of problems, and it is exactly these types of treatments that push the discipline in new directions. As noted, a number of monographs have appeared recently to reintroduce Bayesian inference to a new generation of computer-savvy statisticians. ... However, Gill achieves what these do not: a quality introduction and reference guide to Bayesian inference and MCMC methods that will become a standard in political methodology.
-The Journal of Politics, November 2003

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