Understanding Computational Bayesian

There is a strong upsurge in the use of Bayesian methods in applied statistical analysis, yet most introductory statistics texts only present frequentist methods. In Bayesian statistics the rules of probability are used to make inferences about the parameter. Prior information about the parameter and sample information from the data are combined using Bayes theorem. Bayesian statistics has many important advantages that students should learn about if they are going into fields where statistics will be used. This book uniquely covers the topics usually found in a typical introductory statistics book but from a Bayesian perspective.

William M. Bolstad, Phd, is Senior Lecturer in the Department of Statistics at The University of Waikato (New Zealand). Dr. Bolstad's research interests include Bayesian statistics, MCMC methods, recursive estimation techniques, multiprocess dynamic time series models, and forecasting. He is the author of Introduction to Bayesian Statistics, Second Edition, also published by Wiley.

Preface	p. xi
Introduction to Bayesian Statistics	p. 1
The Frequentist Approach to Statistics	p. 1
The Bayesian Approach to Statistics	p. 3
Comparing Likelihood and Bayesian Approaches to Statistics	p. 6
Computational Bayesian Statistics	p. 19
Purpose and Organization of This Book	p. 20
Monte Carlo Sampling from the Posterior	p. 25
Acceptance-Rejection-Sampling	p. 27
Sampling-Importance-Resampling	p. 33
Adaptive-Rejection-Sampling from a Log-Concave Distribution	p. 35
Why Direct Methods Are Inefficient for High-Dimension Parameter Space	p. 42
Bayesian Inference	p. 47
Bayesian Inference from the Numerical Posterior	p. 47
Bayesian Inference from Posterior Random Sample	p. 54
Bayesian Statistics Using Conjugate Priors	p. 61
One-Dimensional Exponential Family of Densities	p. 61
Distributions for Count Data	p. 62
Distributions for Waiting Times	p. 69
Normally Distributed Observations with Known Variance	p. 75
Normally Distributed Observations with Known Mean	p. 78
Normally Distributed Observations with Unknown Mean and Variance	p. 80
Multivariate Normal Observations with Known Covariance Matrix	p. 85
Observations from Normal Linear Regression Model	p. 87
Appendix: Proof of Poisson Process Theorem	p. 97
Markov Chains	p. 101
Stochastic Processes	p. 102
Markov Chains	p. 103
Time-Invariant Markov Chains with Finite State Space	p. 104
Classification of States of a Markov Chain	p. 109
Sampling from a Markov Chain	p. 114
Time-Reversible Markov Chains and Detailed-Balance	p. 117
Markov Chains with Continuous State Space	p. 120
Markov Chain Monte Carlo Sampling from Posterior	p. 127
Metropolis-Hastings Algorithm for a Single Parameter	p. 130
Metropolis-Hastings Algorithm for Multiple Parameters	p. 137
Blockwise Metropolis Hastings Algorithm	p. 144
Gibbs Sampling	p. 149
Summary	p. 150
Statistical Inference from a Markov Chain Monte Carlo Sample	p. 159
Mixing Properties of the Chain	p. 160
Finding a Heavy-Tailed Matched Curvature Candidate Density	p. 162
Obtaining An Approximate Random Sample For Inference	p. 168
Appendix: Procedure for Finding the Matched Curvature Candidate Density for a Multivariate Parameter	p. 176
Logistic Regression	p. 179
Logistic Regression Model	p. 180
Computational Bayesian Approach to the Logistic Regression Model	p. 184
Modelling with the Multiple Logistic Regression Model	p. 192
Poisson Regression and Proportional Hazards Model	p. 203
Poisson Regression Model	p. 204
Computational Approach to Poisson Regression Model	p. 207
The Proportional Hazards Model	p. 214
Computational Bayesian Approach to Proportional Hazards Model	p. 218
Gibbs Sampling and Hierarchical Models	p. 235
Gibbs Sampling Procedure	p. 236
The Gibbs Sampler for the Normal Distribution	p. 237
Hierarchical Models and Gibbs Sampling	p. 242
Modelling Related Populations with Hierarchical Models	p. 244
Appendix: Proof That Improper Jeffrey's Prior Distribution for the Hypervariance Can Lead to an Improper Posterior	p. 261
Going Forward with Markov Chain Monte Carlo	p. 265
Using the Included Minitab Macros	p. 271
Using the Included R Functions	p. 289
References	p. 307
Topic Index	p. 313
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Understanding Computational Bayesian Statistics

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