SSpprriinnggeerr SSeerriieess iinn SSttaattiissttiiccss AAddvviissoorrss:: PP.. BBiicckkeell,, PP.. DDiiggggllee,, SS.. FFiieennbbeerrgg,, KK.. KKrriicckkeebbeerrgg,, 11.. OOllkkiinn,, NN.. WWeennnnuutthh,, SS.. ZZeeggeerr SSpprriinnggeerr SScciieennccee++BBuussiinneessss MMeeddiiaa,, LLLLCC Springer Series in Statistics AndersenlBorganlGil//Keiding: Statistical Models Based on Counting Processes. AtkinsonlRiani: Robust Diagnotstic Regression Analysis. Berger: Statistical Decision Theory and Bayesian Analysis, 2nd edition. BolfarinelZacks: Prediction Theory for Finite Populations. Borg/Groenen: Modem Multidimensional Scaling: Theory and Applications Brockwell/Davis: Time Series: Theory and Methods, 2nd edition. Chan/Tong: Chaos: A Statistical Perspective. ChenlShao/Ibrahim: Monte Carlo Methods in Bayesian Computation. David/Edwards: Annotated Readings in the History of Statistics. DevroyelLugosi: Combinatorial Methods in Density Estimation. Efromovich: Nonparametric Curve Estimation: Methods, Theory, and Applications. Eggermont/LaRiccia: Maximum Penalized Likelihood Estimation, Volume I: Density Estimation. Fahrmeirlfutz: Multivariate Statistical Modelling Based on Generalized Linear Models, 2nd edition. Farebrother: Fitting Linear Relationships: A History of the Calculus of Observations 1750-1900. Federer: Statistical Design and Analysis for Intercropping Experiments, Volume I: Two Crops. Federer: Statistical Design and Analysis for Intercropping Experiments, Volume II: Three or More Crops. Fienberg/Hoag/inlKruskal/fanur (Eds.): A Statistical Model: Frederick Mosteller's Contributions to Statistics, Science and Public Policy. Fisher/Sen: The Collected Works of Wassily Hoeffding. GlazlNauslWallenstein: Scan Statistics. Good: Permutation Tests: A Practical Guide to Resampling Methods for Testing Hypotheses, 2nd edition. Gourieroux: ARCH Models and Financial Applications. Grandell: Aspects of Risk Theory. Haberman: Advanced Statistics, Volume I: Description of Populations. Hall: The Bootstrap and Edgeworth Expansion. Hardie: Smoothing Techniques: With Implementation in S. Harrell: Regression Modeling Strategies: With Applications to Linear Models, Logistic Regression, and Survival Analysis Hart: Nonparametric Smoothing and Lack-of-Fit Tests. Hartigan: Bayes Theory. HastielfibshiranilFriedman: The Elements of Statistical Leaming: Data Mining, Inference, and Prediction Hedayat/SloanelStujken: Orthogonal Arrays: Theory and Applications. Heyde: Quasi-Likelihood and its Application: A General Approach to Optimal Parameter Estimation. Huet/BouvieriGruet/lolivet: Statistical Tools for Nonlinear Regression: A Practical Guide with S-PLUS Examples. Ibrahim/ChenlSinha: Bayesian Survival Analysis. Kolen/Brennan: Test Equating: Methods and Practices. (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 excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use of general descriptive names, trade names, trademarks, etc., in this publication, even ifthe former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely by anyone. 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