Handbook of Markov Chain Monte Carlo .pdf

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Handbook of Markov Chain Monte Carlo

Chapman & Hall/CRC Handbooks of Modern Statistical Methods Series Editor Garrett Fitzmaurice Department o f Biostatistics Harvard School o f Public Health Boston, MA, U.S.A. Aims and Scope The objective o f the series is to provide high-quality volumes covering the state-of-the-art in the theory and applications o f statistical methodology. The hooks in the series are thoroughly edited and present comprehensive, coherent, and unified summaries o f specific methodological topics from statistics. The chapters are written by the leading researchers in the field, and present a good balance o f theory and application through a synthesis o f the key methodological developments and examples and case studies using real data. The scope o f the series is wide, covering topics o f statistical methodology that are well developed and find application in a range o f scientific disciplines. The volumes are primarily o f interest to researchers and graduate students from statistics and biostatistics, but also appeal to scientists from fields where the methodology is applied to real problems, including medical research, epidemiology and public health, engineering, biological science, environmental science, and the social sciences. Published Titles Longitudinal Data Analysis Edited by Garrett Fitzmaurice, Marie Davidian, Geert Verheke, and Geert Molenherghs Handbook of Spatial Statistics Edited by Alan E, Gelfand, Peter J. Diggle, Montserrat Fuentes, and Peter Guttorp Handbook of M arkov Chain Monte Carlo Edited by Steve Brooks, Andrew Gelman, Galin L. Jones, andXiao-Li Meng

Chapman & Hall/CRC Handbooks of Modern Statistical Methods Handbook of Markov Chain Monte Carlo E dited by Steve Brooks Andrew Gelman Galin L. Jones Xiao-Li Meng CRC Press Taylor S. Francis Group Boca Ralon London New York C R C Press is an im p rin l of the Taylor & Francis Croup, an informa business A C H A P M A N & H AL L BO O K

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Contents Preface................................................................................................................................................... xix Editors...................................................................................................................................................xxi Contributors...................................................................................................................................... xxiii Part I Foundations, Methodology, and Algorithms 1. Introduction to Markov Chain Monte Carlo..........................................................................3 1.10 Variance Estimation Charles J. Geyer 1.1 History....................................................................................................................................3 1.2 Markov Chains.....................................................................................................................4 1.3 Computer Programs and Markov Chains .....................................................................5 1.4 Stationaiity............................................................................................................................ 5 1.5 Reversibility..........................................................................................................................6 1.6 Functionals............................................................................................................................ 6 1.7 The Theory of Ordinary Monte Carlo ...........................................................................6 1.8 The Theory of M CM C........................................................................................................ 8 1.8.1 Multivariate Theory...............................................................................................8 1.8.2 The Autocovariance Function............................................................................. 9 1.9 AR{1) Exam ple.....................................................................................................................9 1.9.1 A Digression, on Toy Problem s...................................................................... 10 1.9.2 Supporting Technical R ep o rt.........................................................................11 1.9.3 The Exam ple...................................................................................................... 11 ........................................................................................................13 1.10.1 Nonoverlapping Batch M ean s...................................................................... 13 Initial Sequence Methods................................................................................ 16 1.10.2 1.10.3 Initial Sequence Methods and Batch M e a n s..............................................17 1 11 The Practice of M C M C ................................................................................................. 17 1.11.1 Black Box MCMC ............................................................................................ 18 1.11.2 Pseudo-Convergence........................................................................................18 1.11.3 One Long Run versus Many Short Runs..................................................... 18 1.11.4 Bu rn-In................................................................................................................ 19 1.11.5 Diagnostics.........................................................................................................21 1.12 Elementary Theory of M CM C..................................................................................... 22 1.12.1 The Metropolis-Hastings Update................................................................. 22 1.12.2 The Metropolis-Hastings Theorem...............................................................23 1.12.3 The Metropolis U p d ate...................................................................................24 1.12.4 The Gibbs U p d ate............................................................................................ 24 1.12.5 Vaiiable-at-a-Time Metropolis-Hastings..................................................... 25 1.12.6 Gibbs Is a Special Case of Metropolis-Hastings .......................................26 1.12.7 Combining Updates.......................................................................................... 26 1.12.7.1 Composition...................................................................................... 26 1.12.7.2 Palindromic Com position..............................................................26 1.12.8 State-Independent M ixing..............................................................................26 1.12.9 Subsampling.......................................................................................................27 1.12.10 Gibbs and Metropolis Revisited.................................................................... 28 v

vi Contents 1.13 A Metropolis Example..................................................................................................... 29 1.14 Checkpointing..................................................................................................................34 1.15 Designing M C M C C ode................................................................................................ 35 1.16 Validating and Debugging MCMC C od e...................................................................36 1.17 The Metropolis-Hasting^ Green Algorithm..............................................................37 1.17.1 State-Dependent Mixing ................................................................................38 1.17.2 Radon-Nikodym Derivatives.........................................................................39 .................................................. 40 1.17.3 Measure-Theoretic Metropolis-Hastings 1.17.3.1 Metropolis-Hastings-Green Elementary U p d ate....................40 1.17.3.2 The MHG Theorem ..........................................................................42 1.17.4 MHG with Jacobians and Augmented State S p a c e .................................45 1.17.4.1 The MHGJ Theorem ........................................................................46 Acknowledgments.................................................................................................................... 47 References...................................................................................................................................47 2. A Short History of MCMC: Subjective Recollections from Incomplete D ata........49 2.2.1 2.2.2 Christian. Rd>ert and George CaseUn 2.1 Introduction.......................................................................................................................49 2.2 Before the Revolution..................................................................................................... 50 The Metropolis et al. (1953) Paper.................................................................50 The Hastings (1970) P ap er............................................................................. 52 2.3 Seeds of the Revolution...................................................................................................53 Besag and the Fundamental (Missing) Theorem ......................................53 2.3.1 2.3.2 EM and Its Simulated Versions as Precursors...........................................53 2.3.3 Gibbs and Beyond.............................................................................................54 2.4 The Revolution..................................................................................................................54 2.4.1 Advances in MCMC Theory...........................................................................56 2.4.2 Advances in MCMC A pplications.............................................................. 57 2.5 After the Revolution........................................................................................................58 2.5.1 A Brief Glimpse atPartide System s............................................................58 2.5.2 Perfect Sam pling...............................................................................................58 2.5.3 Reversible Jump and Variable Dimensions................................................59 2.5.4 Regeneration and the Central Limit Theorem ...........................................59 2.6 Conclusion.........................................................................................................................60 Acknowledgments.................................................................................................................... 61 References...................................................................................................................................61 3. Reversible Jump MCMC..........................................................................................................67 Yimtvi F(m mid Scott A. Sisson 3.1 3.2 Introduction.......................................................................................................................67 3.1.1 From Metropolis-Hastmgs to Reversible Jum p........................................ 67 3.1.2 Application A re a s............................................................................................ 68 Im plementation............................................................................................................... 71 3.2.1 Mapping Functions and Proposal Distributions......................................72 3.2.2 Marginalization and Augmentation............................................................ 73 3.2.3 Centering and Order M ethods......................................................................74 3.2.4 Multi-Step Proposals....................................................................................... 77 3.2.5 Generic Samplers...............................................................................................78

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