Preface

Sampling from the posterior distribution and computing posterior quantities of interest using Markov chain Monte Carlo (MCMC) samples are two major challenges involved in advanced Bayesian computation. This book examines each of these issues in detail and focuses heavily on computing various posterior quantities of interest from a given MCMC sample. Several topics are addressed, including techniques for MCMC sampling, Monte Carlo (MC) methods for estimation of posterior summaries, improving simulation accuracy, marginal posterior density estimation, estimation of normalizing constants, constrained parameter problems, Highest Posterior Density (HPD) interval calculations, computation of posterior modes, and posterior computations for proportional hazards models and Dirichlet process models. Also extensive discussion is given for computations involving model comparisons, including both nested and nonnested models. Marginal likelihood methods, ratios of normalizing constants, Bayes factors, the Savage-Dickey density ratio, Stochastic Search Variable Selection (SSVS), Bayesian Model Averaging (BMA), the reverse jump algorithm, and model adequacy using predictive and latent residual approaches are also discussed.

The book presents an equal mixture of theory and real applications. Theoretical and applied problems are given in Exercises at the end of each chapter. The book is structured so that the methodology and applications are presented in the main body of each chapter and all rigorous proofs and derivations are placed in Appendices. This should enable a wide audience of readers to use the book without having to go through the technical details. Several types of models are used to demonstrate the various computational methods. We discuss generalized linear models, generalized linear mixed models, order restricted models, models for ordinal response data, semiparametric proportional hazards models, and nonparametric models using the Dirichlet process. Each of these models is demonstrated with real data. The applications are mainly from the health sciences, including food science, agriculture, cancer, AIDS, the environment, and education.

The book is intended as a graduate textbook or a reference book for a one-semester course at the advanced Master's or Ph.D. level. The prerequisites include one course in statistical inference and Bayesian theory at the level of Casella and Berger (1990)Casella, G.Berger, R.L. and BoxBox, G.E.P. and TiaoTiao, G.C. (1992). Thus, this book would be most suitable for second- or third-year graduate students in statistics or biostatistics. It would also serve as a useful reference book for applied or theoretical researchers as well as practitioners. Moreover, the book presents several open research problems that could serve as useful thesis topics.

We would like to acknowledge the following people, who have helped us in making this book possible. We thank Alan E. Gelfand for sending us the Table of Contents for his book, Jun S. Liu for his help on Multiple-Try Metropolis algorithms, grouped and collapsed Gibbs, grouped move and multigrid MC sampling, and dynamic weighting algorithms for Chapters 2 and 3, Chuanhai Liu for his help on the CA-adjusted MCMC algorithm, Siddhartha Chib for his suggestions on the Metropolis algorithm, Metropolized Carlin-Chib algorithm, marginal likelihood estimation, and other helpful comments, Man-Suk Oh for her extensions to the IWMDE algorithm, and Linghau Peng and her advisor Edward I. George for sending us the copy of her Ph.D. thesis on normalizing constant estimation for discrete distribution simulation, Dipak K. Dey for many helpful discussions and suggestions, Luke Tierney for helpful comments in the early stages of the book, and Xiao-Li Meng for providing us with several useful papers on estimation of normalizing constants.

We also thank Colleen Lewis of the Department of Mathematical Sciences at Worcester Polytechnic Institute for her editorial assistance. Finally, we owe deep thanks to our families for their constant love, patience, understanding, and support. It is to them that we dedicate this book.

July, 1999 Ming-Hui Chen, Qi-Man Shao, and Joseph G. Ibrahim