Advanced Markov chain Monte Carlo methods

Cover......Page 1
Title Page......Page 6
Copyright......Page 7
Dedication......Page 8
Contents......Page 10
Preface......Page 16
Acknowledgments......Page 20
Publisher’s Acknowledgments......Page 22
1.1 Bayes......Page 24
1.1.2 The Jeffreys Priors and Beyond......Page 25
1.2.1 Credible Intervals and Regions......Page 27
1.2.2 Hypothesis Testing: Bayes Factors......Page 28
1.3.1 The Problem......Page 31
1.3.3 Monte Carlo via Importance Sampling......Page 32
1.4 Random Variable Generation......Page 33
1.4.2 Acceptance-Rejection Methods......Page 34
1.4.3 The Ratio-of-UniformsMethod and Beyond......Page 37
1.5.1 Markov Chains......Page 41
1.5.2 Convergence Results......Page 43
1.5.3 Convergence Diagnostics......Page 46
Exercises......Page 47
2.1 The Gibbs Sampler......Page 50
2.2 Data Augmentation......Page 53
2.3.1 Blocking and Collapsing......Page 56
2.3.2 Hierarchical Centering and Reparameterization......Page 57
2.3.3 Parameter Expansion for Data Augmentation......Page 58
2.3.4 Alternating Subspace-Spanning Resampling......Page 66
2.4.1 The Student-tModel......Page 68
2.4.2 Robit Regression or Binary Regression with the Student-t Link......Page 70
2.4.3 Linear Regression with Interval-Censored Responses......Page 73
Exercises......Page 77
Appendix 2A: The EM and PX-EM Algorithms......Page 79
3.1 The Metropolis-Hastings Algorithm......Page 82
3.1.1 Independence Sampler......Page 85
3.1.3 Problems withMetropolis-Hastings Simulations......Page 86
3.2.2 The Langevin Algorithm......Page 88
3.2.3 The Multiple-Try MH Algorithm......Page 89
3.3.1 Reversible Jump MCMC Algorithm......Page 90
3.3.2 Change-Point Identification......Page 93
3.4.1 Metropolis-Within-Gibbs Sampler......Page 98
3.4.2 Bayesian Analysis for ChIP-chip Data......Page 99
Exercises......Page 106
4 Auxiliary Variable MCMC Methods......Page 108
4.1 Simulated Annealing......Page 109
4.2 Simulated Tempering......Page 111
4.3 The Slice Sampler......Page 113
4.4 The Swendsen-Wang Algorithm......Page 114
4.5 The Wolff Algorithm......Page 116
4.6 The Mo/ller Algorithm......Page 118
4.7 The Exchange Algorithm......Page 120
4.8 The Double MH Sampler......Page 121
4.8.1 Spatial AutologisticModels......Page 122
4.9.1 Monte Carlo MH Algorithm......Page 126
4.9.2 Convergence......Page 130
4.9.3 Spatial AutologisticModels (Revisited)......Page 133
4.9.4 Marginal Inference......Page 134
4.10 Applications......Page 136
4.10.1 AutonormalModels......Page 137
4.10.2 Social Networks......Page 139
Exercises......Page 144
5 Population-Based MCMC Methods......Page 146
5.1 Adaptive Direction Sampling......Page 147
5.2 Conjugate GradientMonte Carlo......Page 148
5.3 Sample Metropolis-Hastings Algorithm......Page 149
5.4 Parallel Tempering......Page 150
5.5 EvolutionaryMonte Carlo......Page 151
5.5.1 Evolutionary Monte Carlo in Binary-Coded Space......Page 152
5.5.2 EvolutionaryMonte Carlo in Continuous Space......Page 155
5.5.3 Implementation Issues......Page 156
5.5.4 Two Illustrative Examples......Page 157
5.5.5 Discussion......Page 162
5.6 Sequential Parallel Tempering for Simulation of High Dimensional Systems......Page 163
5.6.1 Build-up Ladder Construction......Page 164
5.6.3 An Illustrative Example: the Witch’s Hat Distribution......Page 165
5.6.4 Discussion......Page 168
5.7 Equi-Energy Sampler......Page 169
5.8.1 Bayesian Curve Fitting......Page 171
5.8.2 Protein Folding Simulations: 2D HPModel......Page 176
5.8.3 Bayesian Neural Networks for Nonlinear Time Series Forecasting......Page 179
Exercises......Page 185
Appendix 5A: Protein Sequences for 2D HPModels......Page 186
6.1.1 The IWIWPrinciple......Page 188
6.1.2 Tempering Dynamic Weighting Algorithm......Page 190
6.1.3 DynamicWeighting in Optimization......Page 194
6.2.1 The Basic Idea......Page 196
6.2.2 A Theory of DWIS......Page 197
6.2.3 Some IWIWp Transition Rules......Page 199
6.2.4 Two DWIS Schemes......Page 202
6.2.5 Weight Behavior Analysis......Page 203
6.2.6 A Numerical Example......Page 206
6.3.1 Sampling from Distributions with Intractable Normalizing Constants......Page 208
6.3.2 Monte Carlo Dynamically Weighted Importance Sampling......Page 209
6.3.3 Bayesian Analysis for Spatial Autologistic Models......Page 214
6.4 Sequentially Dynamically Weighted Importance Sampling......Page 218
Exercises......Page 220
7 Stochastic Approximation Monte Carlo......Page 222
7.1 MulticanonicalMonte Carlo......Page 223
7.2 1/k-Ensemble Sampling......Page 225
7.3 The Wang-Landau Algorithm......Page 227
7.4 Stochastic ApproximationMonte Carlo......Page 230
7.5.1 Efficient p-Value Evaluation for Resampling- Based Tests......Page 241
7.5.2 Bayesian Phylogeny Inference......Page 245
7.5.3 Bayesian Network Learning......Page 250
7.6.1 Smoothing SAMC forModel Selection Problems......Page 256
7.6.2 Continuous SAMC for Marginal Density Estimation......Page 262
7.6.3 Annealing SAMC for Global Optimization......Page 267
7.7.1 Convergence......Page 276
7.7.2 Convergence Rate......Page 290
7.7.3 Ergodicity and its IWIWProperty......Page 294
7.8 Trajectory Averaging: Toward the Optimal Convergence Rate......Page 298
7.8.1 Trajectory Averaging for a SAMCMC Algorithm......Page 300
7.8.2 Trajectory Averaging for SAMC......Page 302
7.8.3 Proof of Theorems 7.8.2 and 7.8.3.......Page 304
Exercises......Page 319
Appendix 7A: Test Functions for Global Optimization......Page 321
8 Markov Chain Monte Carlo with Adaptive Proposals......Page 328
8.1 Stochastic Approximation-Based Adaptive Algorithms......Page 329
8.1.1 Ergodicity and Weak Law of Large Numbers......Page 330
8.1.2 Adaptive Metropolis Algorithms......Page 332
8.2 Adaptive Independent Metropolis-Hastings Algorithms......Page 335
8.3.1 Identification of Regeneration Times......Page 338
8.4.1 ADS, EMC, NKC andMore......Page 340
8.4.2 Adaptive EMC......Page 341
8.4.3 Application to Sensor Placement Problems......Page 346
Exercises......Page 347
References......Page 350
Index......Page 376