Bayesian Methods

PREFACES

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

MONTE CARLO 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: RR@R 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

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

ADVANCED MARKOV CHAIN MONTE CARLO
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

APPENDIX C: INTRODUCTION TO THE BUGS LANGUAGE
General Process
Technical Background on the Algorithm
WinBUGS Features
JAGS Programming

REFERENCES

AUTHOR INDEX
SUBJECT INDEX