Markov Chain Monte Carlo in Practice

Transcription

1 Markov Chain Monte Carlo in Practice

2 Markov Chain Monte Carlo in Practice Edited by W.R. Gilks Medical Research Council Biostatistics Unit Cambridge UK S. Richardson French National Institute for Health and Medical Research Vilejuif France and D.J. Spiegelhalter Medical Research Council Biostatistics Unit Cambridge UK I unl SPRlNGER-SCIENCE+BUSINESS MEDIA, B.V.

3 First edition 1996 Springer Science+Business Media Dordrecht 1996 Originally published by Chapman & Hall in 1996 Softcover reprint of the hardcover 1 st edition 1996 ISBN DOI / ISBN (ebook) Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the UK Copyright Designs and Patents Act, 1988, this publication may not be reproduced, stored, or transmitted, in any form or by any means, without the prior permission in writing of the publishers, or in the case of reprographic reproduction only in accordance with the terms of the licences issued by the Copyright Licensing Agency in the UK, or in accordance with the terms of licences issued by the appropriate Reproduction Rights Organization outside the UK. Enquiries concerning reproduction outside the terms stated here should be sent to the publishers at the London address printed on this page. The publisher makes no representation, express or implied, with regard to the accuracy of the information contained in this book and cannot accept any legal responsibility or liability for any errors or omissions that may be made. A Catalogue record for this book is available from the British Library

4 Contents Contributors xv 1 Introducing Markov chain Monte Carlo 1 W. R. Gilks, S. Richardson and D. J. Spiegelhalter 1.1 Introduction The problem Bayesian inference Calculating expectations Markov chain Monte Carlo Monte Carlo integration Markov chains The Metropolis-Hastings algorithm Implementation Canonical forms of proposal distribution Blocking Updating order Number of chains Starting values Determining burn-in Determining stopping time Output analysis Discussion 16 2 Hepatitis B: a case study in MCMC methods 21 D. J. Spiegelhalter, N. G. Best, W. R. Gilks and H. Inskip 2.1 Introduction Hepatitis B immunization Background Preliminary analysis Modelling Structural modelling Probability modelling Prior distributions 27

5 vi CONTENTS Fitting a model using Gibbs sampling Initialization Sampling from full conditional distributions Monitoring the output Inference from the output Assessing goodness-of-fit 2.5 Model elaboration Heavy-tailed distributions Introducing a covariate 2.6 Conclusion Appendix: BUGS Markov chain concepts related to sampling algorithms G. O. Roberts 3.1 Introduction 3.2 Markov chains 3.3 Rates of convergence 3.4 Estimation Batch means Window estimators 3.5 The Gibbs sampler and Metropolis-Hastings algorithm The Gibbs sampler The Metropolis-Hastings algorithm Introduction to general state-space Markov chain theory L. Tierney 4.1 Introduction 4.2 Notation and definitions 4.3 Irreducibility, recurrence and convergence Irreducibility Recurrence Convergence 4.4 Harris recurrence 4.5 Mixing rates and central limit theorems 4.6 Regeneration 4.7 Discussion Full conditional distributions W. R. Gilks 5.1 Introduction 5.2 Deriving full conditional distributions A simple example Graphical models 5.3 Sampling from full conditional distributions

6 CONTENTS vii Rejection sampling Ratio-of-uniforms method Adaptive rejection sampling Metropolis-Hastings algorithm Hybrid adaptive rejection and Metropolis-Hastings Discussion 86 6 Strategies for improving MCMC 89 W. R. Gilks and G. O. Roberts 6.1 Introduction Reparameterization Correlations and transformations Linear regression models Random-effects models Nonlinear models General comments on reparameterization Random and adaptive direction sampling The hit-and-run algorithm Adaptive direction sampling (ADS) Modifying the stationary distribution Importance sampling Metropolis-coupled MCMC Simulated tempering Auxiliary variables Methods based on continuous-time processes Discussion IInplementing MCMC 115 A. E. Raftery and S. M. Lewis 7.1 Introduction Determining the number of iterations Software and implementation Output analysis An example Generic Metropolis algorithms An example Discussion Inference and monitoring convergence 131 A. Gelman 8.1 Difficulties in inference from Markov chain simulation The risk of undiagnosed slow convergence Multiple sequences and overdispersed starting points Monitoring convergence using simulation output 136

7 viii 8.5 Output analysis for inference 8.6 Output analysis for improving efficiency CONTENTS Model determination using sampling-based methods 145 A. E. Gelfand 9.1 Introduction Classical approaches The Bayesian perspective and the Bayes factor Alternative predictive distributions Cross-validation predictive densities Posterior predictive densities Other predictive densities How to use predictive distributions Computational issues Estimating predictive densities Computing expectations over predictive densities Sampling from predictive densities An example Discussion Hypothesis testing and model selection 163 A. E. Raftery 10.1 Introduction Uses of Bayes factors Marginal likelihood estimation by importance sampling Marginal likelihood estimation using maximum likelihood The Laplace-Metropolis estimator Candidate's estimator The data-augmentation estimator Application: how many components in a mixture? Gibbs sampling for Gaussian mixtures A simulated example How many disks in the Galaxy? Discussion 181 Appendix: S-PLUS code for the Laplace-Metropolis estimator Model checking and model improvement 189 A. Gelman and X.-L. Meng 11.1 Introduction Model checking using posterior predictive simulation Model improvement via expansion Example: hierarchical mixture modelling of reaction times The data and the basic model Model checking using posterior predictive simulation 196

8 CONTENTS ix Expanding the model Checking the new model Stochastic search variable selection 203 E. I. George and R. E. McCulloch 12.1 Introduction A hierarchical Bayesian model for variable selection Searching the posterior by Gibbs sampling Extensions SSVS for generalized linear models SSVS across exchangeable regressions Constructing stock portfolios with SSVS ~. Discussion Bayesian model comparison via jump diffusions 215 D. B. Phillips and A. F. M. Smith 13.1 Introduction Model choice Example 1: mixture deconvolution Example 2: object recognition Example 3: variable selection in regression Example 4: change-point identification Jump-diffusion sampling The jump component Moving between jumps Mixture deconvolution Dataset 1: galaxy velocities Dataset 2: length of porgies Object recognition Results Variable selection Change-point identification Dataset 1: Nile discharge Dataset 2: facial image Conclusions Estimation and optimization of functions 241 C. J. Geyer 14.1 Non-Bayesian applications of MCMC Monte Carlo optimization Monte Carlo likelihood analysis Normalizing-constant families Missing data Decision theory 251

9 x CONTENTS 14.7 Which sampling distribution? 14.8 Importance sampling 14.9 Discussion Stochastic EM: method and application 259 J. Diebolt and E. H. S. Jp 15.1 Introduction The EM algorithm The stochastic EM algorithm Stochastic imputation Looking at the plausible region Point estimation Variance of the estimates Examples Type-I censored data Empirical Bayes probit regression for cognitive diagnosis Generalized linear mixed models 275 D. G. Clayton 16.1 Introduction Generalized linear models (GLMs) Bayesian estimation of GLMs Gibbs sampling for GLMs Generalized linear mixed models (GLMMs) Frequentist GLMMs Bayesian GLMMs Specification of random-effect distributions Prior precision Prior means Intrinsic aliasing and contrasts Autocorrelated random effects The first-difference prior The second-difference prior General Markov random field priors Interactions Hyperpriors and the estimation of hyperparameters Some examples Longitudinal studies Time trends for disease incidence and mortality Disease maps and ecological analysis Simultaneous variation in space and time Frailty models in survival analysis Discussion 298

10 CONTENTS xi 17 Hierarchical longitudinal modelling 303 B. P. Carlin 17.1 Introduction Clinical background Model detail and MCMC implementation Results Summary and discussion Medical monitoring 321 C. Berzuini 18.1 Introduction Modelling medical monitoring Nomenclature and data Linear growth model Marker growth as a stochastic process Computing posterior distributions Recursive updating Forecasting Model criticism Illustrative application The clinical problem The model Parameter estimates Predicting deterioration Discussion MCMC for nonlinear hierarchical models 339 J. E. Bennett, A. Racine-Poon and J. C. Wakefield 19.1 Introduction Implementing MCMC Method 1: Rejection Gibbs Method 2: Ratio Gibbs Method 3: Random-walk Metropolis Method 4: Independence Metropolis-Hastings Method 5: MLE/prior Metropolis-Hastings Comparison of strategies Guinea pigs data A case study from pharmacokinetics-pharmacodynamics Extensions and discussion Bayesian mapping of disease 359 A. Mollie 20.1 Introduction Hypotheses and notation 360

11 xii CONTENTS 20.3 Maximum likelihood estimation of relative risks Hierarchical Bayesian model of relative risks Bayesian inference for relative risks Specification of the prior distribution Graphical representation of the model Empirical Bayes estimation of relative risks The conjugate gamma prior Non-conjugate priors Disadvantages of EB estimation Fully Bayesian estimation of relative risks Choices for hyperpriors Full conditional distributions for Gibbs sampling Example: gall-bladder and bile-duct cancer mortality Discussion MCMC in image analysis 381 P. J. Green 21.1 Introduction The relevance of MCMC to image analysis Image models at different levels Pixel-level models Pixel-based modelling in SPECT Template models An example of template modelling Stochastic geometry models Hierarchical modelling Methodological innovations in MCMC stimulated by imaging Discussion Measurement error 401 S. Richardson 22.1 Introduction Conditional-independence modelling Designs with individual-level surrogates Designs using ancillary risk-factor information Estimation Illustrative examples Two measuring instruments with no validation group Influence of the exposure model Ancillary risk-factor information and expert coding Discussion Gibbs sampling methods in genetics 419 D. C. Thomas and W. J. Gauderman

12 CONTENTS xiii 23.1 Introduction Standard methods in genetics Genetic terminology Genetic models Genetic likelihoods Gibbs sampling approaches Gibbs sampling of genotypes Gibbs sampling of parameters Initialization, convergence, and fine tuning MCMC maximum likelihood Application to a family study of breast cancer Conclusions Mixtures of distributions: inference and estimation 441 C. P. Robert 24.1 Introduction Modelling via mixtures A first example: character recognition Estimation methods Bayesian estimation The missing data structure Gibbs sampling implementation General algorithm Extra-binomial variation Normal mixtures: star clustering Reparameterization issues Extra-binomial variation: continued Convergence of the algorithm Testing for mixtures Extra-binomial variation: continued Infinite mixtures and other extensions Dirichlet process priors and nonparametric models Hidden Markov models An archaeological example: radiocarbon dating 465 C. Litton and C. Buck 25.1 Introduction Background to radiocarbon dating Archaeological problems and questions Illustrative examples Example 1: dating settlements Example 2: dating archaeological phases Example 3: accommodating outliers Practical considerations 477

13 xiv CONTENTS Index 25.5 Discussion

14 Contributors James E Bennett Carlo Berzuini Nicola G Best Caitlin Buck Bradley P Carlin David G Clayton Jean Diebolt Department of Mathematics, Imperial College, London, UK. Dipartimento di Informatica e Sistemistica, University of Pavia, Italy. Medical Research Council Biostatistics Unit, Institute of Public Health, Cambridge, UK. School of History and Archaeology, University of Wales, Cardiff, UK. Division of Biostatistics, School of Public Health, University of Minnesota, Minneapolis, USA. Medical Research Council Biostatistics Unit, Institute of Public Health, Cambridge, UK. Departement de Statistique et Modeles AIeatoires, CNRS, Universite Paris 6, France. W James Gauderman Department of Preventive Medicine, University of Southern California, Los Angeles, USA. Alan E Gelfand Andrew Gelman Edward I George Department of Statistics, University of Connecticut, USA. Department of Statistics, University of California, Berkeley, California, USA. MSIS Department, University of Texas at Austin, USA.

15 xvi CONTRIBUTORS Charles J Geyer Walter R Gilks Peter J Green Hazel Inskip Eddie H S Ip Steven M Lewis Cliff Litton Robert E McCulloch Xiao-Li Meng Annie Mollie David B Phillips Amy Racine-Poon Adrian E Raftery Sylvia Richardson Christian PRobert Gareth 0 Roberts Adrian F M Smith School of Statistics, University of Minnesota, Minneapolis, USA. Medical Research Council Biostatistics Unit, Institute of Public Health, Cambridge, UK. Department of Mathematics, University of Bristol, UK. Medical Research Council Environmental Epidemiology Unit, Southampton, UK. Educational Testing Service, Princeton, USA. School of Social Work, University of Washington, Seattle, USA. Department of Mathematics, University of Nottingham, UK. Graduate School of Business, University of Chicago, USA. Department of Statistics, University of Chicago, USA. INSERM Unite 351, Villejuif, France. NatWest Markets, London, UK. Ciba Geigy, Basle, Switzerland. Department of Statistics, University of Washington, Seattle, USA. INSERM Unite 170, Villejuif, France. Laboratoire de Statistique, University of Rouen, France. Statistical Laboratory, University of Cambridge, UK. Department of Mathematics, Imperial College, London, UK. David J Spiegelhalter Medical Research Council Biostatistics Unit, Institute of Public Health, Cambridge, UK.

16 CONTRIBUTORS Duncan C Thomas Luke Tierney Jon C Wakefield Department of Preventive Medicine, University of Southern California, Los Angeles, USA. School of Statistics, University of Minnesota, Minneapolis, USA. xvii Department of Mathematics, Imperial College, London, UK.