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Handbook of Markov Chain Monte Carlo
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Preface X1,Editors xxi,Contributors xxiii,Part I Foundations Methodology and Algorithms. 1 Introduction to Markov Chain Monte Carlo 3,Charles J Geyer. 1 1 History 3,1 2 Markov Chains 4,1 3 Computer Programs and Markov Chains 5. 1 4 Stationarity 5,1 5 Reversibility 6,1 6 Functionals 6. 1 7 The Theory of Ordinary Monte Carlo 6,1 8 The Theory of MCMC 8.
1 8 1 Multivariate Theory 8,1 8 2 The Autocovariance Function 9. 1 9 AR 1 Example 9,1 9 1 A Digression on Toy Problems 10. 1 9 2 Supporting Technical Report 11,1 9 3 The Example 11. 1 10 Variance Estimation 13,1 10 1 Nonoverlapping Batch Means 13. 1 10 2 Initial Sequence Methods 16, 1 10 3 Initial Sequence Methods and Batch Means 17.
1 11 The Practice of MCMC 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 Burn In 19,1 11 5 Diagnostics 21,1 12 Elementary Theory of MCMC 22. 1 12 1 The Metropolis Hastings Update 22,1 12 2 The Metropolis Hastings Theorem 23. 1 12 3 The Metropolis Update 24,1 12 4 The Gibbs Update 24.
1 12 5 Variable 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 PalindromicComposition 26,1 12 8 State Independent Mixing 26. Subsampling,1 12 10 Gibbs and Metropolis Revisited 28. vi Contents,1 13 A Metropolis Example 29,Checkpointing.
1 15 Designing MCMC Code 35,1 16 Validating and Debugging MCMC Code 36. 1 17 The Metropolis Hastings Green Algorithm 37,1 17 1 State Dependent Mixing 38. Derivatives 39,1 17 2 Radon Nikodym,1 17 3 Measure Theoretic Metropolis Hastings 40. 1 17 3 1 Metropolis Hastings Green Elementary Update. 1 17 3 2 The MHG Theorem 42, 1 17 4 MHG with Jacobians and Augmented State Space 45. 1 17 4 1 The MHGJ Theorem 46,Acknowledgments,References 47.
2 A Short History of MCMC Subjective Recollections from Incomplete Data 49. Christian Robert and George Casella,2 1 Introduction 49. 2 2 Before the Revolution 50,2 2 1 The Metropolis et al 1953 Paper 50. 2 2 2 The Hastings 1970 Paper 52,2 3 Seeds of the Revolution 53. 2 3 1 and the Fundamental Missing Theorem 53, 2 3 2 EM and Its Simulated Versions as Precursors 53. 2 3 3 Gibbs and Beyond,2 4 The Revolution 54,2 4 1 Advances in MCMC Theory 56.
2 4 2 Advances in MCMC Applications 57,2 5 After the Revolution 58. A Brief at Particle Systems 58,2 5 1 Glimpse,2 5 2 Perfect 58. 2 5 3 Reversible Jump and Variable Dimensions 59,2 5 4 Regeneration and the Central Limit Theorem. 2 6 Conclusion 60,Acknowledgments 61,References 61. 3 Reversible Jump MCMC 67,Yanan Fan and Scott A Sisson.
3 1 Introduction 67,3 1 1 From Metropolis Hastings to Reversible Jump. 3 1 2 Application Areas,Implementation,Functions and Distributions 72. 3 2 1 Mapping Proposal,3 2 2 Marginalization and Augmentation 73. 3 2 3 Centering and Order Methods 74,3 2 4 Multi Step Proposals 77. 3 2 5 Generic Samplers,Contents vn,3 3 Post Simulation 80.
3 3 1 Label Switching 80,3 3 2 Convergence Assessment 81. 3 3 3 Estimating Bayes Factors 82,3 4 Related Multi Model Sampling Methods 84. 3 4 1 Jump Diffusion 84,3 4 2 Product Space Formulations 85. 3 4 3 Point Process Formulations 85,3 4 4 Multi Model Optimization 85. 3 4 5 Population MCMC 86,3 4 6 Multi Model Sequential Monte Carlo 86.
3 5 Discussion and Future Directions 86,Acknowledgments 87. References 87, 4 Optimal Proposal Distributions and Adaptive MCMC 93. Jeffrey S Rosenthal,4 1 Introduction 93,4 1 1 TheMetropolis Hastings Algorithm 93. 4 1 2 Optimal Scaling 93,4 1 3 Adaptive MCMC 94,4 1 4 Comparing Markov Chains 94. 4 2 Optimal Scaling of Random Walk Metropolis,4 2 1 Basic Principles 95.
4 2 2 Rate d 96,Optimal Acceptance as oo,4 2 3 Inhomogeneous Target Distributions. 4 2 4 Metropolis Adjusted Langevin Algorithm,4 2 5 Numerical Examples. 4 2 5 1 Off Diagonal Covariance 100,4 2 5 2 Inhomogeneous Covariance 100. 4 2 6 Frequently Asked Questions,4 3 Adaptive MCMC. 4 3 1 Ergodicity of Adaptive MCMC,4 3 2 Adaptive Metropolis.
4 3 3 Adaptive Metropolis within Gibbs,4 3 4 State Dependent Proposal Scalings. 4 3 5 Limit Theorems 107,Frequently Asked Questions. 4 4 Conclusion 109,References 110,5 MCMC Using Hamiltonian Dynamics 113. Radford M Neal,5 1 Introduction 113,5 2 Hamiltonian Dynamics 114. 5 2 1 Hamilton s Equations 114,5 2 1 1 Equations of Motion 114.
5 2 1 2 Potential and Kinetic Energy 115,5 2 1 3 A One Dimensional Example 116. viii Contents,5 2 2 Properties of Hamiltonian Dynamics 116. 5 2 2 1 Reversibility 116,5 2 2 2 Conservation of the Hamiltonian 116. 5 2 2 3 Volume Preservation 117,5 2 2 4 Symplecticness 119. 5 2 3 Discretizing Hamilton s Equations The Leapfrog Method 119. 5 2 3 1 Euler s Method 119,5 2 3 2 A Modification of Euler s Method 121.
5 2 3 3 The Leapfrog Method 121, 5 2 3 4 Local and Global Error of Discretization Methods 122. 5 3 MCMC from Hamiltonian Dynamics 122, 5 3 1 Probability and the Hamiltonian Canonical Distributions 122. 5 3 2 The Hamiltonian Monte Carlo Algorithm 123,5 3 2 1 The Two Steps of the HMC Algorithm 124. 5 3 2 2 Proof That HMC Leaves the Canonical,Distribution Invariant 126. 5 3 2 3 Ergodicity of HMC 127,5 3 3 Illustrations of HMC and Its Benefits 127.
5 3 3 1 Trajectories for a Two Dimensional Problem 127. 5 3 3 2 Sampling from a Two Dimensional Distribution 128. 5 3 3 3 The Benefit of Avoiding Random Walks 130, 5 3 3 4 Sampling from a 100 Dimensional Distribution 130. 5 4 HMC in Practice and Theory 133,5 4 1 Effect of Linear Transformations 133. 5 4 2 Tuning HMC 134,5 4 2 1 Preliminary Runs and Trace Plots 134. 5 4 2 2 What Stepsize 135,5 4 2 3 What Trajectory Length 137. 5 4 2 4 Using Multiple Stepsizes 137,5 4 3 Combining HMC with Other MCMC Updates 138.
5 4 4 Scaling with Dimensionality 139, 5 4 4 1 Creating Distributions of Increasing Dimensionality. by Replication 139, 5 4 4 2 Scaling of HMC and Random Walk Metropolis 139. 5 4 4 3 Optimal Acceptance Rates 141, 5 4 4 4 Exploring the Distribution of Potential Energy 142. 5 4 5 HMC for Hierarchical Models 142,5 5 Extensions of and Variations on HMC 144. 5 5 1 Discretization by Splitting Handling Constraints and Other. Applications 145,5 5 1 1 Splitting the Hamiltonian 145.
5 5 1 2 Splitting to Exploit Partial Analytical Solutions 146. 5 5 1 3 Splitting Potential Energies with Variable Computation. 5 5 1 4 Splitting According to Data Subsets 147,5 5 1 5 Handling Constraints 148. 5 5 2 Taking One Step at a Time The Langevin Method 148. 5 5 3 Partial Momentum Refreshment Another Way to Avoid. Random Walks 150,Contents he,5 5 4Acceptance Using Windows of States 152. 5 5 5Using Approximations to Compute the Trajectory 155. 5 5 6 Short Cut Trajectories Adapting the Stepsize without Adaptation. 5 5 7 Tempering during a Trajectory 157,Acknowledgment 160. References 160, 6 Inference from Simulations and Monitoring Convergence 163. Andrew Gelman and Kenneth Shirley,6 1 Quick Summary of Recommendations 163.
6 2 Key Differences between Point Estimation and MCMC Inference 164. 6 3 Inference for Functions of the Parameters vs Inference for Functions of the. Target Distribution 166,6 4 Inference from Noniterative Simulations 167. 6 5 Burn In 168, 6 6 Monitoring Convergence Comparing between and within Chains 170. 6 7 Inference from Simulations after Approximate Convergence 171. 6 8 Summary 172,Acknowledgments 173,References 173. 7 Implementing MCMC Estimating with Confidence 175. James M Flegal and Galin L Jones,7 1 Introduction 175. 7 2 Initial Examination of Output 176,7 3 Point Estimates of 6 178.
7 3 1 Expectations 178,7 3 2 Quantiles 181,7 4 Interval Estimates of 0 182. 7 4 1 Expectations 182,7 4 1 1 Overlapping Batch Means 182. 7 4 1 2 Parallel Chains 184,7 4 2 Functions of Moments 185. 7 4 3 Quantiles 187,7 4 3 1 Subsampling Bootstrap 187. 7 4 4 Multivariate Estimation 189,7 5 Estimating Marginal Densities 189.
7 6 Terminating the Simulation 192,7 7 Markov Chain Central Limit Theorems 193. 7 8 Discussion 194,Acknowledgments,References 195,8 Perfection within Reach Exact MCMC Sampling 199. Radu V Craiu and Xiao Li Meng,8 1 Intended Readership 199. 8 2 Coupling from the Past 199, 8 2 1 Moving from Time Forward to Time Backward 199. x Contents,8 2 2 Hitting the Limit 200,8 2 3 Challenges for Routine Applications 201.
8 3 Coalescence Assessment 201,8 3 1 Illustrating MonotoneCoupling 201. 8 3 2 Illustrating Brute Force Coupling 202,8 3 3 General Classes of Monotone Coupling 203. 8 3 4 Bounding Chains 204, 8 4 Cost Saving Strategies for Implementing Perfect Sampling 206. 8 4 1 Read Once CFTP 206,8 4 2 Fill s Algorithm 208. 8 5 Coupling Methods 210,8 5 1 Splitting Technique 211.
8 5 2 Coupling via a Common Proposal 212,8 5 3 Coupling via Discrete Data Augmentation 213. 8 5 4 Perfect Slice Sampling 215,8 6 Swindles 217, 8 6 1 Efficient Use of Exact Samples via Concatenation 218. 8 6 2 Multistage Perfect Sampling 219,8 6 3 Antithetic Perfect Sampling 220. Integrating Exact and Approximate MCMC Algorithms 221. 8 7 Where Are the Applications 223,Acknowledgments 223. References 223,9 Spatial Point Processes 227,Mark Ruber.
9 1 Introduction 227,9 2 Setup 227, 9 3 Metropolis Hastings Reversible Jump Chains 230. 9 3 1 Examples 232,9 3 2 Convergence 232, 9 4 Continuous Time Spatial Birth Death Chains 233. 9 4 1 Examples 235, 9 4 2 Shifting Moves with Spatial Birth and Death Chains 236. 9 4 3 Convergence 236,9 5 Perfect Sampling 236,9 5 1 Acceptance Rejection Method 236. 9 5 2 Dominated Coupling from the Past 238,9 5 3 Examples 242.
9 6 Monte Carlo Posterior Draws 243,9 7 Running Time Analysis 245. 9 7 1 Running Time of Perfect Simulation Methods 248. Acknowledgment 251,References 251, 10 The Data Augmentation Algorithm Theory and Methodology 253. James P Hobert,10 1 Basic Ideas and,Examples 253,Contents xi. 10 2 Properties of the DA Markov Chain 261,10 2 1 BasicRegularity Conditions 261. 10 2 2 BasicConvergence Properties 263,10 2 3 Geometric Ergodicity 264.
10 2 4 Central Limit Theorems 267,10 3 Choosing the Monte Carlo Sample Size 269. 10 3 1 Classical Monte Carlo 269, 10 3 2 Three Markov Chains Closely Related to X 270. 10 3 3 Minorization Regeneration and an Alternative CLT 272. 10 3 4 Simulating the Split Chain 275, 10 3 5 A General Method for Constructing the Minorization Condition. 10 4 Improving the DA Algorithm 279, 10 4 1 The PX DA and Marginal Augmentation Algorithms 280. 10 4 2 The Operator Associated with a Reversible Markov Chain 284. 10 4 3 A Theoretical Comparison of the DA and PX DA Algorithms 286. 10 4 4 Is There a Best PX DA Algorithm 288,Acknowledgments 291.
References 291, 11 Importance Sampling Simulated Tempering and Umbrella Sampling 295. Charles Geyer,11 1 Importance Sampling 295,11 2 Simulated Tempering 297. 11 2 1 Parallel Tempering Update 299,11 2 2 Serial. Tempering Update 300,11 2 3 Effectiveness of Tempering 300. 11 2 4 Tuning Serial Tempering 301,11 2 5 Umbrella Sampling 302.
11 3 Bayes Factors and Normalizing Constants 303,11 3 1 Theory 303. 11 3 2 Practice 305,11 3 2 1 Setup 305,11 3 2 2 Trial and Error 307. 11 3 2 3 Monte Carlo Approximation 308,11 3 3 Discussion 309. Acknowledgments 310,References 310,12 Likelihood Free MCMC 313. Scott A Sisson and Yanan Fan,12 1 Introduction 313.
12 2 Review of Likelihood Free Theory and Methods 314. 12 2 1 Likelihood Free Basics 314, 12 2 2 The Nature of the Posterior Approximation 315. 12 2 3 A Simple Example 316,12 3 Likelihood Free MCMC Samplers 317. 12 3 1 Marginal Space Samplers 319,12 3 2 Error Distribution Augmented Samplers 320. xii Contents,12 3 3 Potential Alternative MCMC Samplers 321. 12 4 A Practical Guide to Likelihood Free MCMC 322. 12 4 1 An Exploratory Analysis 322,12 4 2 The Effect of 324.
12 4 3 The Effect of the Weighting Density 326,12 4 4 The Choice of Summary Statistics 327. 12 4 5 Improving Mixing 329,12 4 6 Evaluating Model Misspecification 330. 12 5 Discussion 331,Acknowledgments 333,References 333. Part II Applications and Case Studies, 13 MCMC in the Analysis of Genetic Data on Related Individuals 339. Elizabeth Thompson,13 1 Introduction 339, 13 2 Pedigrees Genetic Variants and the Inheritance of Genome 340.
13 3 Conditional Independence Structures of Genetic Data 341. 13 3 1 Genotypic Structure of Pedigree Data 342,13 3 2 Inheritance Structure of Genetic Data 344. 13 3 3 Identicalby Descent Structure of Genetic Data 347. 13 3 4 ibd Graph Computations for Markers and Traits 348. 13 4 MCMC Sampling of Latent Variables 349,Genotypes and Meioses 349. 13 4 2 Some Block Gibbs Samplers 349, 13 4 3 Gibbs Updates and Restricted Updates on Larger Blocks 350. 13 5 MCMC Sampling of Inheritance Given Marker Data 351. 13 5 1 Sampling Inheritance Conditional on Marker Data 351. 13 5 2 Monte Carlo EM and Likelihood Ratio Estimation 351. 13 5 3 Importance Sampling Reweighting 353, 13 6 Using MCMC Realizations for Complex Trait Inference 354. 13 6 1 Estimating a Likelihood Ratio or lod Score 354. 13 6 2 Uncertainty in Inheritance and Tests for,Linkage Detection 356.
13 6 3 Localization of Causal Loci Using Latent p Values 357. 13 7 Summary 358,Acknowledgment 359,References 359. 14 An MCMC Based Analysis of a Multilevel Model for Functional MRI Data. Brian Caffo DuBois Bowman Lynn Eberly and Susan Spear Bassett. 14 1 Introduction 363,14 1 1 Literature Review 364. 14 1 2 Example Data 365, 14 2 Data Preprocessing and First Level Analysis 367. 14 3 A Multilevel Model for Incorporating Regional Connectivity 368. 14 3 1 Model 368,14 3 2 Simulating the Markov Chain. 14 4 Analyzing the Chain,14 4 1 Activation Results.
14 5 Connectivity Results,14 5 1 Intra Regional Connectivity. 14 5 2 Inter Regional Connectivity,14 6 Discussion. References, 15 Partially Collapsed Gibbs Sampling and Path Adaptive. Metropolis Hastings in High Energy Astrophysics,David A van Dyk and Taeyonng Park. 15 1 Introduction,15 2 Partially Collapsed Gibbs Sampler.
15 3 Path Adaptive Metropolis Hastings Sampler,15 4 Spectra Analysis in High Energy Astrophysics. 15 5 Efficient MCMC in Spectral Analysis,15 6 Conclusion. Acknowledgments,References, 16 Posterior Exploration for Computationally Intensive Forward Models. David Higdon C Shane Reese David Moulton Jasper A Vrugt and Colin Fox. 16 1 Introduction 401, 16 2 An Inverse Problem in Electrical Impedance Tomography. 16 2 1 Exploration via,Posterior Single Site Metropolis Updates.
16 3 Multivariate Updating Schemes,16 3 1 Random Walk Metropolis. 16 3 2 Differential Evolution and Variants,16 4 Augmenting with Fast Approximate Simulators. 16 4 1 Delayed Acceptance Metropolis,16 4 2 An Augmented Sampler. 16 5 Discussion, Appendix Formulation Based on a Process Convolution Prior. Acknowledgments,References,17 Statistical Ecology,17 1 Introduction.
17 2 Analysis of Ring Recovery Data,17 2 1 Covariate Analysis. 17 2 1 1 Posterior Conditional Distributions,17 2 1 2 Results. 17 2 2 Mixed Effects Model,17 2 2 1 Obtaining Posterior Inference. 17 2 2 2 Posterior Conditional Distributions,17 2 2 3 Results. xiv Contents,17 2 3 Model Uncertainty 428,17 2 3 1 Model Specification 430.
17 2 3 2 Reversible jump Algorithm 430,17 2 3 3 Proposal Distribution 431. 17 2 3 4 Results 431,17 2 3 5 Comments 432,17 3 Analysis of Count Data 433. 17 3 1 State Space Models 434,17 3 1 1 System Process 434. 17 3 1 2 Observation Process 434,17 3 1 3 Model 435. 17 3 1 4 Obtaining Inference 435,17 3 2 Integrated Analysis 435.
17 3 2 1 MCMC Algorithm 436,17 3 2 2 Results 437,17 3 3 Model Selection 439. 17 3 3 1 Results 440,17 3 3 2 Comments 442,17 4 Discussion 444. References 445, 18 Gaussian Random Field Models for Spatial Data 449. Murali Haran,18 1 Introduction 449,18 1 1 Some Motivation for Spatial Modeling 449. 18 1 2 MCMC and Spatial Models A Shared History 451. 18 2 Linear Spatial Models 451,18 2 1 Linear Gaussian Process Models 452.
18 2 1 1 MCMC for Linear GPs 453, 18 2 2 Linear Gaussian Markov Random Field Models 454. 18 2 2 1 MCMC for Linear GMRFs 457,18 2 3 Summary 457. 18 3 Spatial Generalized Linear Models 458,18 3 1 The Generalized Linear Model Framework 458. 18 3 2 Examples 459,18 3 2 1 Binary Data 459,18 3 2 2 Count Data 460. 18 3 2 3 Zero Inflated Data 462,18 3 3 MCMC for SGLMs 463.
18 3 3 1 Langevin Hastings MCMC 463, 18 3 3 2 Approximating an SGLM by a Linear Spatial Model 465. 18 3 4 Maximum Likelihood Inference for SGLMs 467,18 3 5 Summary 467. 18 4 Non Gaussian Markov Random Field Models 468,18 5 Extensions 470. 18 6 Conclusion 471,Acknowledgments 473,References 473. Contents xv, 19 Modeling Preference Changes via a Hidden Markov Item Response.
Theory Model 479,Jong Hee Park,19 1 Introduction 479. 19 2 Dynamic Ideal Point Estimation 480,19 3 Hidden Markov Item Response Theory Model 481. 19 4 Preference 487,Changes in US Supreme Court Justices. 19 5 Conclusions 490,Acknowledgments,References 490. 20 Parallel Bayesian MCMC Imputation for Multiple Distributed. Models A Case Study in Environmental Epidemiology 493. Brian Caffo Roger Peng Francesca Dominici Thomas A Louis. and Scott Zeger,20 1 Introduction 493,20 2 The Data Set 494.
Bayesian Imputation,20 3 1 Single Lag Models,20 3 2 Distributed Lag Models 496. 20 4 Model and Notation 498, 20 4 1 Prior and Hierarchical Model Specification 501. Bayesian Imputation,20 5 1 501,20 5 2 A Parallel Imputation Algorithm 502. 20 6 of the Medicare Data 504,Appendix Full Conditionals 509. Acknowledgment 510,References 510,21 MCMC for Models 513.
State Space,Paul Fearnhead,21 1 Introduction State Space Models. 21 2 Bayesian Analysis and MCMC Framework 515,the State 515. 21 3 Updating,21 3 1 Single Site Updates of the State 515. 21 3 2 Block Updates for the State 518,21 3 3 Other Approaches 523. the Parameters 523,21 4 Updating,21 4 1 Conditional Updates of the Parameters 523.
21 4 2 Reparameterization of the Model, 21 4 3 Joint Updates of the Parameters and State 526. 21 5 Discussion 527,References 527,xvi Contents,22 MCMC in Educational Research 531. Robert Mislevy and John T Behrens,22 1 Introduction 531. 22 2 Statistical Models in Education Research 532,22 3 Historical and Current Research Activity. 22 3 1 Multilevel Models 534,22 3 2 Psychometric Modeling 535.
22 3 2 1 Continuous Latent and Observable Variables 535. 22 3 2 2 Continuous Latent Variables and Discrete Observable. Variables 536,22 3 2 3 Discrete Latent Variables and Discrete. Observable Variables 537,22 3 2 4 Combinations of Models 538. 22 4 NAEP Example 538,22 5 Discussion Advantages of MCMC 541. 22 6 Conclusion 542,References 542,23 Applications of MCMC in Fisheries Science. Russell B Millar,23 1 Background 547,23 2 The Current Situation 549.
23 2 1 Software 550,23 2 2 of MCMC in Fisheries 551. Perception,23 3 ADMB 551,23 3 1 Automatic Differentiation 551. 23 3 2 Metropolis Hastings Implementation 552,23 4 Bayesian Applications to Fisheries. 23 4 1 553,Capturing Uncertainty,23 4 1 1 State Space Models of South Atlantic. Albacore Tuna Biomass 553,23 4 1 2Implementation 555.
23 4 2 Hierarchical Modeling of Research Trawl,Catchability 555. 23 4 3 Hierarchical Modeling of Stock Recruitment,Relationship 557. 23 5 Concluding Remarks,Acknowledgment 561,References 561. 24 Model Comparison and Simulation for Hierarchical Models. Analyzing Rural Urban Migration in Thailand 563,Filiz Garip and Bruce Western. 24 1 Introduction 563,24 2 Thai Migration Data 564.
Results 568,24 3 Regression,24 4 Posterior Predictive Checks 569. Contents xvii, 24 5 Exploring Model Implications with Simulation 570.

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