Monte Carlo Methods in Bayesian Computation

398 Pages·2000·14.338 MB·English

by  Ming-Hui Chen, Qi-Man Shao, Joseph G. Ibrahim (auth.)

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(continued after index)
Ming-Hui Chen
Qi-Man Shao
Joseph G. Ibrahim
Monte Carlo Methods
in Bayesian Computation
With 32 Illustrations
i
Springer
Ming-Hui Chen Qi-Man Shao
Department of Mathematical Sciences Department of Mathematics
Worcester Polytechnic Institute University of Oregon
Worcester, MA 01609-2280 Eugene, OR 97403-1222
USA USA
[email protected] [email protected]
Joseph G. Ibrahim
Department of Biostatistics
Harvard School of Public Health
and Dana-Farber Cancer Institute
Boston, MA 02115
USA
[email protected]
Library of Congress Cataloging-in-Publication Data
Chen, Ming-Hui, 1961-
Monte Carlo methods in Bayesian computation / Ming-Hui Chen, Qi-Man Shao, Joseph
G.lbrahim
p. cm. - (Springer series in statistics)
Inc\udes bibliographical references and indexes.
ISBN 978-1-4612-7074-4 ISBN 978-1-4612-1276-8 (eBook)
DOI 10.1007/978-I -46 12-1276-8
1. Bayesian statistical decision theory. 2. Monte Carlo method. 1. Shao, Qi-Man.
II. Ibrahim, Joseph George. III. Title. IV. Series.
QA279.5.C57 2000
519.5'42--{jc21 99-046366
Printed on acid-free paper.
© 2000 Springer Science+Business Media New York
Originally published by Springer-Verlag New York, Inc. in 2000
Softcover reprint of the hardcover 1s I edition 2000
AII rights reserved. This work may not be translated or copied in whole or in par! without the
written permission of the publisher (Springer Science+Business Media, LLC) except for brief
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Production managed by A. Orrantia; manufacturing supervised by Jerome Basma.
Camera-ready copy prepared from the authors' LaTeX files.
9 8 7 6 5 4 3 2 (Corrected second printing, 2002)
ISBN 978-1-4612-7074-4
To Lan Bai, Jiena Miao, and Mona Ibrahim
Victoria, Paula, and Wenqi
Preface
Sampling from the posterior distribution and computing posterior quanti
ties 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 comput
ing 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, improv
ing simulation accuracy, marginal posterior density estimation, estimation
of normalizing constants, constrained parameter problems, Highest Poste
rior 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 in
volving model comparisons, including both nested and nonnested models.
Marginal likelihood methods, ratios of normalizing constants, Bayes fac
tors, 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 audi
ence of readers to use the book without having to go through the technical
details. Several types of models are used to demonstrate the various compu-
viii Preface
tational methods. We discuss generalized linear models, generalized linear
mixed models, order restricted models, models for ordinal response data,
semiparametric proportional hazards models, and non parametric 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 prereq
uisites include one course in statistical inference and Bayesian theory at
the level of Casella and Berger (1990) and Box and Tiao (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-'fry
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 algo
rithm, 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 send
ing 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, under
standing, and support. It is to them that we dedicate this book.
July, 1999 Ming-Hui Chen, Qi-Man Shao, and Joseph G. Ibrahim
Contents
Preface vii
1 Introduction 1
1.1 Aims .. 1
1.2 Outline. 2
1.3 Motivating Examples 4
1.4 The Bayesian Paradigm. 14
Exercises ........ . 15
2 Markov Chain Monte Carlo Sampling 19
2.1 Gibbs Sampler ......... . 20
2.2 Metropolis-Hastings Algorithm .. 23
2.3 Hit-and-Run Algorithm ...... . 26
2.4 Multiple-Try Metropolis Algorithm 30
2.5 Grouping, Collapsing, and Reparameterizations 31
2.5.1 Grouped and Collapsed Gibbs ..... . 31
2.5.2 Reparameterizations: Hierarchical Centering and
Rescaling. . . . . . . . . . . . . . . . . . . . . 34
2.5.3 Collapsing and Reparameterization for Ordinal Re-
sponse Models . . . . . . . . . . . . . . . . . . .. 35
2.5.4 Hierarchical Centering for Poisson Random Effects
Models. . . . . . . . . . . . . . . . . . 40
2.6 Acceleration Algorithms for MCMC Sampling . . . . .. 42
x Contents
2.6.1 Grouped Move and Multigrid Monte Carlo Sam-
pling . . . . . . . . . . . . . . . . . . . . 43
2.6.2 Covariance-Adjusted MCMC Algorithm 45
2.6.3 An Illustration ..... 47
2.7 Dynamic Weighting Algorithm. 52
2.8 Toward "Black-Box" Sampling. 55
2.9 Convergence Diagnostics 59
Exercises ........ . 63
3 Basic Monte Carlo Methods for Estimating Posterior
Quantities 67
3.1 Posterior Quantities . . . . . . . . . . . 67
3.2 Basic Monte Carlo Methods . . . . . . 68
3.3 Simulation Standard Error Estimation 71
3.3.1 Time Series Approach .... 71
3.3.2 Overlapping Batch Statistics .. 73
3.4 Improving Monte Carlo Estimates ... 77
3.4.1 Variance-Reduction MCMC Sampling. 77
3.4.2 Weighted Monte Carlo Estimates 81
3.5 Controlling Simulation Errors 86
Appendix. 89
Exercises ........... . 89
4 Estimating Marginal Posterior Densities 94
4.1 Marginal Posterior Densities 95
4.2 Kernel Methods .. . 97
4.3 IWMDE Methods ..... . 98
4.4 Illustrative Examples . . . . 104
4.5 Performance Study Using the Kullback-Leibler Divergence 107
Appendix. 117
Exercises . . . . . . . . . . . . . . . . . . . . 121
5 Estimating Ratios of Normalizing Constants 124
5.1 Introduction............... 124
5.2 Importance Sampling . . . . . . . . . . 125
5.2.1 Importance Sampling-Version 1 126
5.2.2 Importance Sampling~Version 2 126
5.3 Bridge Sampling. . . . . . . . . . 127
5.4 Path Sampling. . . . . . . . . . . . 129
5.4.1 Univariate Path Sampling . 130
5.4.2 Multivariate Path Sampling 131
5.4.3 Connection Between Path Sampling and Bridge
Sampling . . . . . . . 132
5.5 Ratio Importance Sampling 132
5.5.1 The Method. . . . . 132
Contents xi
5.5.2 Implementation.......... 135
5.6 A Theoretical Illustration. . . . . . . . . 138
5.7 Computing Simulation Standard Errors. 143
5.8 Extensions to Densities with Different Dimensions. 145
5.8.1 Why Different Dimensions? ......... 145
5.8.2 General Formulation . . . . . . . . . . . . . 146
5.8.3 Extensions of the Previous Monte Carlo Methods 147
5.8.4 Global Optimal Estimators. . . . . . . . . . . .. 149
5.8.5 Implementation Issues . . . . . . . . . . . . . .. 152
5.9 Estimation of Normalizing Constants After Transformation 160
5.10 Other Methods ............. 161
5.10.1 Marginal Likelihood Approach. . 162
5.10.2 Reverse Logistic Regression ... 163
5.10.3 The Savage-Dickey Density Ratio 164
5.11 An Application of Weighted Monte Carlo Estimators 165
5.12 Discussion 172
Appendix. 173
Exercises. 187
6 Monte Carlo Methods for Constrained Parameter Prob
lems 191
6.1 Constrained Parameter Problems . . . . . . . . . . . . . 191
6.2 Posterior Moments and Marginal Posterior Densities .. 193
6.3 Computing Normalizing Constants for Bayesian Estimation 196
6.4 Applications . . . . . . . . . . . . . . . . . . . . 199
6.4.1 The Meal, Ready-to-Eat (MRE) Model. 199
6.4.2 Job Satisfaction Example. 205
6.5 Discussion 208
Appendix. 208
Exercises . 211
7 Computing Bayesian Credible and HPD Intervals 213
7.1 Bayesian Credible and HPD Intervals. 214
7.2 Estimating Bayesian Credible Intervals 216
7.3 Estimating Bayesian HPD Intervals . . 218
7.3.1 An Order Statistics Approach 219
7.3.2 Weighted Monte Carlo Estimation of HPD Intervals 220
7.4 Extension to the Constrained Parameter Problems 221
7.5 Numerical Illustration ............... . 223
7.5.1 A Simulation Study ............ . 223
7.5.2 Meal, Ready-to-Eat (MRE) Data Example 224
7.6 Discussion 226
Appendix. 227
Exercises. 232