Bayesian Methods for Data Analysis

538 Pages·2008·4.37 MB·English

by  Bradley P. Carlin, Thomas A. Louis

Bayesian Methods
for Data Analysis
Third Edition
C6978_FM.indd 1 5/7/08 2:04:37 PM
CHAPMAN & HALL/CRC
Texts in Statistical Science Series
Series Editors
Bradley P. Carlin, University of Minnesota, USA
Julian J. Faraway, University of Bath, UK
Martin Tanner, Northwestern University, USA
Jim Zidek, University of British Columbia, Canada
Analysis of Failure and Survival Data Epidemiology — Study Design and
P. J. Smith Data Analysis, Second Edition
M. Woodward
The Analysis of Time Series—
An Introduction, Sixth Edition Essential Statistics, Fourth Edition
C. Chatfield D.A.G. Rees
Applied Bayesian Forecasting and Time Series Extending the Linear Model with R:
Analysis Generalized Linear, Mixed Effects and
A. Pole, M. West and J. Harrison Nonparametric Regression Models
J.J. Faraway
Applied Nonparametric Statistical Methods,
Fourth Edition A First Course in Linear Model Theory
P. Sprent and N.C. Smeeton N. Ravishanker and D.K. Dey
Applied Statistics — Handbook of GENSTAT Generalized Additive Models:
Analysis An Introduction with R
E.J. Snell and H. Simpson S. Wood
Applied Statistics — Principles and Examples Interpreting Data — A First Course
D.R. Cox and E.J. Snell in Statistics
A.J.B. Anderson
Bayesian Data Analysis, Second Edition
An Introduction to Generalized
A. Gelman, J.B. Carlin, H.S. Stern
Linear Models, Third Edition
and D.B. Rubin
A.J. Dobson and A.G. Barnett
Bayesian Methods for Data Analysis,
Introduction to Multivariate Analysis
Third Edition
C. Chatfield and A.J. Collins
B.P. Carlin and T.A. Louis
Introduction to Optimization Methods and
Beyond ANOVA — Basics of Applied
Their Applications in Statistics
Statistics
B.S. Everitt
R.G. Miller, Jr.
Introduction to Probability with R
Computer-Aided Multivariate Analysis, K. Baclawski
Fourth Edition
Introduction to Randomized Controlled
A.A. Afifi and V.A. Clark
Clinical Trials, Second Edition
A Course in Categorical Data Analysis J.N.S. Matthews
T. Leonard
Introduction to Statistical Methods for
A Course in Large Sample Theory Clinical Trials
T.S. Ferguson Thomas D. Cook and David L. DeMets
Data Driven Statistical Methods Large Sample Methods in Statistics
P. Sprent P.K. Sen and J. da Motta Singer
Decision Analysis — A Bayesian Approach Linear Models with R
J.Q. Smith J.J. Faraway
Elementary Applications of Probability Markov Chain Monte Carlo —
Theory, Second Edition Stochastic Simulation for Bayesian
H.C. Tuckwell Inference, Second Edition
Elements of Simulation D. Gamerman and H.F. Lopes
B.J.T. Morgan Mathematical Statistics
K. Knight
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Texts in Statistical Science
Bayesian Methods
for Data Analysis
Third Edition
Bradley P. Carlin
Univesity of Minnesota
Minneapolis, MN, U.S.A.
Thomas A. Louis
Johns Hopkins Bloomberg School of Public Health
Baltimore, MD, U.S.A.
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Chapman & Hall/CRC
Taylor & Francis Group
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Library of Congress Cataloging-in-Publication Data
Carlin, Bradley P.
Bayesian methods for data analysis / authors, Bradley P. Carlin and Thomas A.
Louis. -- 3rd ed.
p. cm. -- (Chapman & Hall/CRC texts in statistical science
series ; 78)
Originally published: Bayes and Empirical Bayes methods for data analysis. 1st ed.
Includes bibliographical references and index.
ISBN 978-1-58488-697-6 (alk. paper)
1. Bayesian statistical decision theory. I. Louis, Thomas A., 1944- II. Carlin, Bradley
P. Bayes and Empirical Bayes methods for data analysis. III. Title. IV. Series.
QA279.5.C36 2008
519.5’42--dc22 2008019143
Visit the Taylor & Francis Web site at
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to
Caroline, Samuel, Joshua, and Nathan
and
Germaine, Margit, and Erica

Contents
Preface to the Third Edition xiii
1 Approaches for statistical inference 1
1.1 Introduction 1
1.2 Motivating vignettes 2
1.2.1 Personalprobability 2
1.2.2 Missing data 2
1.2.3 Bioassay 3
1.2.4 Attenuation adjustment 4
1.3 Defining the approaches 4
1.4 The Bayes-frequentistcontroversy 6
1.5 Some basic Bayesianmodels 10
1.5.1 A Gaussian/Gaussian (normal/normal)model 11
1.5.2 A beta/binomial model 11
1.6 Exercises 13
2 The Bayes approach 15
2.1 Introduction 15
2.2 Prior distributions 27
2.2.1 Elicited priors 28
2.2.2 Conjugate priors 32
2.2.3 Noninformative priors 36
2.2.4 Other prior construction methods 40
2.3 Bayesian inference 41
2.3.1 Point estimation 41
2.3.2 Interval estimation 48
2.3.3 Hypothesis testing and Bayes factors 50
2.4 Hierarchical modeling 59
2.4.1 Normal linear models 59
2.4.2 Effective model size and the DIC criterion 70
2.5 Model assessment 79
2.5.1 Diagnostic measures 79
viii CONTENTS
2.5.2 Model averaging 89
2.6 Nonparametric methods 93
2.7 Exercises 98
3 Bayesian computation 105
3.1 Introduction 105
3.2 Asymptotic methods 108
3.2.1 Normal approximation 108
3.2.2 Laplace’s method 110
3.3 Noniterative Monte Carlo methods 112
3.3.1 Direct sampling 112
3.3.2 Indirect methods 115
3.4 Markov chain Monte Carlo methods 120
3.4.1 Gibbs sampler 121
3.4.2 Metropolis-Hastings algorithm 130
3.4.3 Slice sampler 139
3.4.4 Hybridforms,adaptiveMCMC,andotheralgorithms 140
3.4.5 Variance estimation 150
3.4.6 Convergence monitoring and diagnosis 152
3.5 Exercises 159
4 Model criticism and selection 167
4.1 Bayesian modeling 168
4.1.1 Linear models 168
4.1.2 Nonlinear models 174
4.1.3 Binary data models 176
4.2 Bayesian robustness 181
4.2.1 Sensitivity analysis 181
4.2.2 Prior partitioning 188
4.3 Model assessment 194
4.4 Bayes factors via marginal density estimation 196
4.4.1 Direct methods 197
4.4.2 Using Gibbs sampler output 198
4.4.3 Using Metropolis-Hastings output 200
4.5 Bayes factors via sampling over the model space 201
4.5.1 Product space search 203
4.5.2 “Metropolized” product space search 205
4.5.3 Reversible jump MCMC 206
4.5.4 Using partial analytic structure 208
4.6 Other model selection methods 210
4.6.1 Penalized likelihood criteria: AIC, BIC, and DIC 210
4.6.2 Predictive model selection 215
4.7 Exercises 217
CONTENTS ix
5 The empirical Bayes approach 225
5.1 Introduction 225
5.2 Parametric EB (PEB) point estimation 226
5.2.1 Gaussian/Gaussianmodels 227
5.2.2 Computation via the EM algorithm 228
5.2.3 EB performance of the PEB 234
5.2.4 Stein estimation 236
5.3 Nonparametric EB (NPEB) point estimation 240
5.3.1 Compound sampling models 240
5.3.2 Simple NPEB (Robbins’ method) 240
5.4 Interval estimation 244
5.4.1 Morris’ approach 245
5.4.2 Marginal posterior approach 246
5.4.3 Bias correction approach 248
5.5 Bayesian processing and performance 251
5.5.1 Univariate stretching with a two-point prior 251
5.5.2 Multivariate Gaussian model 252
5.6 Frequentist performance 253
5.6.1 Gaussian/Gaussianmodel 254
5.6.2 Beta/binomial model 255
5.7 Empirical Bayes performance 258
5.7.1 Point estimation 259
5.7.2 Interval estimation 262
5.8 Exercises 265
6 Bayesian design 269
6.1 Principles of design 269
6.1.1 Bayesiandesign for frequentist analysis 269
6.1.2 Bayesiandesign for Bayesian analysis 271
6.2 Bayesian clinical trial design 274
6.2.1 Classical versus Bayesian trial design 275
6.2.2 Bayesianassurance 277
6.2.3 Bayesianindifference zone methods 279
6.2.4 Other Bayesianapproaches 282
6.2.5 Extensions 286
6.3 Applications in drug and medical device trials 287
6.3.1 Binary endpoint drug trial 287
6.3.2 Cox regressiondevice trial with interim analysis 297
6.4 Exercises 308
7 Special methods and models 311
7.1 Estimating histograms and ranks 311
7.1.1 Bayesianranking 311
7.1.2 Histogram and triple goal estimates 324