Ebook
Advanced Markov Chain Monte Carlo Methods: Learning from Past SamplesISBN: 9781119956808
374 pages
July 2011

Description
Key Features:
 Expanded coverage of the stochastic approximation Monte Carlo and dynamic weighting algorithms that are essentially immune to local trap problems.
 A detailed discussion of the Monte Carlo MetropolisHastings algorithm that can be used for sampling from distributions with intractable normalizing constants.
 Uptodate accounts of recent developments of the Gibbs sampler.
 Comprehensive overviews of the populationbased MCMC algorithms and the MCMC algorithms with adaptive proposals.
This book can be used as a textbook or a reference book for a onesemester graduate course in statistics, computational biology, engineering, and computer sciences. Applied or theoretical researchers will also find this book beneficial.
Table of Contents
Preface xiii
Acknowledgments xvii
Publisher’s Acknowledgments xix
1 Bayesian Inference and Markov Chain Monte Carlo 1
1.1 Bayes 1
1.1.1 Specification of Bayesian Models 2
1.1.2 The Jeffreys Priors and Beyond 2
1.2 Bayes Output 4
1.2.1 Credible Intervals and Regions 4
1.2.2 Hypothesis Testing: Bayes Factors 5
1.3 Monte Carlo Integration 8
1.3.1 The Problem 8
1.3.2 Monte Carlo Approximation 9
1.3.3 Monte Carlo via Importance Sampling 9
1.4 Random Variable Generation 10
1.4.1 Direct or Transformation Methods 1
1.4.2 AcceptanceRejection Methods 11
1.4.3 The RatioofUniforms Method and Beyond 14
1.4.4 Adaptive Rejection Sampling 18
1.4.5 Perfect Sampling 18
1.5 Markov Chain Monte Carlo 18
1.5.1 Markov Chains 18
1.5.2 Convergence Results 20
1.5.3 Convergence Diagnostics 23
Exercises 24
2 The Gibbs Sampler 27
2.1 The Gibbs Sampler 27
2.2 Data Augmentation 30
2.3 Implementation Strategies and Acceleration Methods 33
2.3.1 Blocking and Collapsing 33
2.3.2 Hierarchical Centering and Reparameterization 34
2.3.3 Parameter Expansion for Data Augmentation 35
2.3.4 Alternating SubspaceSpanning Resampling 43
2.4 Applications 45
2.4.1 The Studentt Model 45
2.4.2 Robit Regression or Binary Regression with the Studentt Link 47
2.4.3 Linear Regression with IntervalCensored Responses 50
Exercises 54
Appendix 2A: The EM and PXEM Algorithms 56
3 The MetropolisHastings Algorithm 59
3.1 The MetropolisHastings Algorithm 59
3.1.1 Independence Sampler 62
3.1.2 Random Walk Chains 63
3.1.3 Problems with MetropolisHastings Simulations 63
3.2 Variants of the MetropolisHastings Algorithm 65
3.2.1 The HitandRun Algorithm. 65
3.2.2 The Langevin Algorithm 65
3.2.3 The MultipleTry MH Algorithm 66
3.3 Reversible Jump MCMC Algorithm for Bayesian Model Selection Problems 67
3.3.1 Reversible Jump MCMC Algorithm 67
3.3.2 ChangePoint Identification 70
3.4 MetropolisWithinGibbs Sampler for ChIPchip Data Analysis 75
3.4.1 MetropolisWithinGibbs Sampler 75
3.4.2 Bayesian Analysis for ChIPchip Data 76
Exercises 83
4 Auxiliary Variable MCMC Methods 85
4.1 Simulated Annealing 86
4.2 Simulated Tempering 88
4.3 The Slice Sampler 90
4.4 The SwendsenWang Algorithm 91
4.5 The Wolff Algorithm 93
4.6 The Mo/ller Algorithm 95
4.7 The Exchange Algorithm 97
4.8 The Double MH Sampler 98
4.8.1 Spatial Autologistic Models 99
4.9 Monte Carlo MH Sampler 103
4.9.1 Monte Carlo MH Algorithm 103
4.9.2 Convergence 107
4.9.3 Spatial Autologistic Models (Revisited) 110
4.9.4 Marginal Inference 111
4.10 Applications 113
4.10.1 Autonormal Models 114
4.10.2 Social Networks 116
Exercises 121
5 PopulationBased MCMC Methods 123
5.1 Adaptive Direction Sampling 124
5.2 Conjugate Gradient Monte Carlo 125
5.3 Sample MetropolisHastings Algorithm 126
5.4 Parallel Tempering 127
5.5 Evolutionary Monte Carlo 128
5.5.1 Evolutionary Monte Carlo in BinaryCoded Space 129
5.5.2 Evolutionary Monte Carlo in Continuous Space 132
5.5.3 Implementation Issues 133
5.5.4 Two Illustrative Examples 134
5.5.5 Discussion 139
5.6 Sequential Parallel Tempering for Simulation of High Dimensional Systems 140
5.6.1 Buildup Ladder Construction 141
5.6.2 Sequential Parallel Tempering 142
5.6.3 An Illustrative Example: the Witch’s Hat Distribution 142
5.6.4 Discussion 145
5.7 EquiEnergy Sampler 146
5.8 Applications 148
5.8.1 Bayesian Curve Fitting 148
5.8.2 Protein Folding Simulations: 2D HP Model 153
5.8.3 Bayesian Neural Networks for Nonlinear Time Series Forecasting 156
Exercises 162
Appendix 5A: Protein Sequences for 2D HP Models 163
6 Dynamic Weighting 165
6.1 Dynamic Weighting 165
6.1.1 The IWIW Principle 165
6.1.2 Tempering Dynamic Weighting Algorithm 167
6.1.3 Dynamic Weighting in Optimization 171
6.2 Dynamically Weighted Importance Sampling 173
6.2.1 The Basic Idea 173
6.2.2 A Theory of DWIS 174
6.2.3 Some IWIWp Transition Rules 176
6.2.4 Two DWIS Schemes 179
6.2.5 Weight Behavior Analysis 180
6.2.6 A Numerical Example 183
6.3 Monte Carlo Dynamically Weighted Importance Sampling 185
6.3.1 Sampling from Distributions with Intractable Normalizing Constants 185
6.3.2 Monte Carlo Dynamically Weighted Importance Sampling 186
6.3.3 Bayesian Analysis for Spatial Autologistic Models 191
6.4 Sequentially Dynamically Weighted Importance Sampling 195
Exercises 197
7 Stochastic Approximation Monte Carlo 199
7.1 Multicanonical Monte Carlo 200
7.2 1/kEnsemble Sampling 202
7.3 The WangLandau Algorithm 204
7.4 Stochastic Approximation Monte Carlo 207
7.5 Applications of Stochastic Approximation Monte Carlo 218
7.5.1 Efficient pValue Evaluation for ResamplingBased Tests 218
7.5.2 Bayesian Phylogeny Inference 222
7.5.3 Bayesian Network Learning 227
7.6 Variants of Stochastic Approximation Monte Carlo 233
7.6.1 Smoothing SAMC for Model Selection Problems 233
7.6.2 Continuous SAMC for Marginal Density Estimation 239
7.6.3 Annealing SAMC for Global Optimization 244
7.7 Theory of Stochastic Approximation Monte Carlo 253
7.7.1 Convergence 253
7.7.2 Convergence Rate 267
7.7.3 Ergodicity and its IWIW Property 271
7.8 Trajectory Averaging: Toward the Optimal Convergence Rate 275
7.8.1 Trajectory Averaging for a SAMCMC Algorithm 277
7.8.2 Trajectory Averaging for SAMC 279
7.8.3 Proof of Theorems 7.8.2 and 7.8.3 281
Exercises 296
Appendix 7A: Test Functions for Global Optimization 298
8 Markov Chain Monte Carlo with Adaptive Proposals 305
8.1 Stochastic ApproximationBased Adaptive Algorithms 306
8.1.1 Ergodicity and Weak Law of Large Numbers 307
8.1.2 Adaptive Metropolis Algorithms 309
8.2 Adaptive Independent MetropolisHastings Algorithms 312
8.3 RegenerationBased Adaptive Algorithms 315
8.3.1 Identification of Regeneration Times 315
8.3.2 Proposal Adaptation at Regeneration Times 317
8.4 PopulationBased Adaptive Algorithms 317
8.4.1 ADS, EMC, NKC and More 317
8.4.2 Adaptive EMC 318
8.4.3 Application to Sensor Placement Problems 323
Exercises 324
References 327
Index 353
Reviews
“The book is suitable as a textbook for onesemester courses on Monte Carlo methods, offered at the advance postgraduate levels.” (Mathematical Reviews, 1 December 2012)
"Researchers working in the field of applied statistics will profit from this easytoaccess presentation. Further illustration is done by discussing interesting examples and relevant applications. The valuable reference list includes technical reports which are hard to and by searching in public data bases." (Zentralblatt MATH, 2011)"This book can be used as a textbook or a reference book for a onesemester graduate course in statistics, computational biology, engineering, and computer sciences. Applied or theoretical researchers will also find this book beneficial." (Breitbart.com: Business Wire , 1 February 2011)
"The Markov Chain Monte Carlo method has now become the dominant methodology for solving many classes of computational problems in science and technology." (SciTech Book News, December 2010) 