Markov chain Monte Carlo : stochastic simulation for Bayesian inference.

Format:

Book

Author/Creator:

Gamerman, Dani.

Contributor:

Lopes, Hedibert Freitas.

Series:

Texts in statistical science ; v. 68.

Texts in statistical science

Language:

English

Subjects (All):

Bayesian statistical decision theory.

Markov processes.

Monte Carlo method.

Physical Description:

xvii, 323 pages : illustrations ; 24 cm.

Edition:

Second edition / Dani Gamerman, Hedibert F. Lopes.

Place of Publication:

Boca Raton : Taylor & Francis, 2006.

Summary:

Incorporating changes in theory and highlighting new applications, Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition presents a concise, accessible, and comprehensive introduction to the methods of this valuable simulation technique. The second edition includes access to an Internet site that provides the code, written in R and WinBUGS, used in many of the previously existing and new examples and exercises. More importantly, the self-explanatory nature of the codes will enable modification of the inputs to the codes and variation on many directions will be available for further exploration.

The self-contained text units make MCMC accessible to scientists in other disciplines as well as statisticians. The book will appeal to everyone working with MCMC techniques, especially research and graduate statisticians and biostatisticians, and scientists handling data and formulating models. The book has been substantially reinforced as a first reading of material on MCMC and, consequently, as a textbook for modern Bayesian computation and Bayesian inference courses.

Contents:

1 Stochastic simulation 9

1.2 Generation of discrete random quantities 10

1.2.1 Bernoulli distribution 11

1.2.2 Binomial distribution 11

1.2.3 Geometric and negative binomial distribution 12

1.2.4 Poisson distribution 12

1.3 Generation of continuous random quantities 13

1.3.1 Probability integral transform 13

1.3.2 Bivariate techniques 14

1.3.3 Methods based on mixtures 17

1.4 Generation of random vectors and matrices 20

1.4.1 Multivariate normal distribution 21

1.4.2 Wishart distribution 23

1.4.3 Multivariate Student's t distribution 24

1.5 Resampling methods 25

1.5.1 Rejection method 25

1.5.2 Weighted resampling method 30

1.5.3 Adaptive rejection method 32

2 Bayesian inference 41

2.2 Bayes' theorem 41

2.2.1 Prior, posterior and predictive distributions 42

2.2.2 Summarizing the information 47

2.3 Conjugate distributions 49

2.3.1 Conjugate distributions for the exponential family 51

2.3.2 Conjugacy and regression models 55

2.3.3 Conditional conjugacy 58

2.4 Hierarchical models 60

2.5 Dynamic models 63

2.5.1 Sequential inference 64

2.5.2 Smoothing 65

2.5.3 Extensions 67

2.6 Spatial models 68

2.7 Model comparison 72

3 Approximate methods of inference 81

3.2 Asymptotic approximations 82

3.2.1 Normal approximations 83

3.2.2 Mode calculation 86

3.2.3 Standard Laplace approximation 88

3.2.4 Exponential form Laplace approximations 90

3.3 Approximations by Gaussian quadrature 93

3.4 Monte Carlo integration 95

3.5 Methods based on stochastic simulation 98

3.5.1 Bayes' theorem via the rejection method 100

3.5.2 Bayes' theorem via weighted resampling 101

3.5.3 Application to dynamic models 104

4 Markov chains 113

4.2 Definition and transition probabilities 114

4.3 Decomposition of the state space 118

4.4 Stationary distributions 121

4.5 Limiting theorems 124

4.6 Reversible chains 127

4.7 Continuous state spaces 129

4.7.1 Transition kernels 129

4.7.2 Stationarity and limiting results 131

4.8 Simulation of a Markov chain 132

4.9 Data augmentation or substitution sampling 135

5 Gibbs sampling 141

5.2 Definition and properties 142

5.3 Implementation and optimization 148

5.3.1 Forming the sample 148

5.3.2 Scanning strategies 150

5.3.3 Using the sample 151

5.3.4 Reparametrization 152

5.3.5 Blocking 155

5.3.6 Sampling from the full conditional distributions 156

5.4 Convergence diagnostics 157

5.4.1 Rate of convergence 158

5.4.2 Informal convergence monitors 159

5.4.3 Convergence prescription 161

5.4.4 Formal convergence methods 164

5.5 Applications 169

5.5.1 Hierarchical models 169

5.5.2 Dynamic models 172

5.5.3 Spatial models 176

5.6 MCMC-based software for Bayesian modeling 178

Appendix 5.A BUGS code for Example 5.7 182

Appendix 5.B BUGS code for Example 5.8 184

6 Metropolis-Hastings algorithms 191

6.2 Definition and properties 193

6.3 Special cases 198

6.3.1 Symmetric chains 198

6.3.2 Random walk chains 198

6.3.3 Independence chains 199

6.3.4 Other forms 204

6.4 Hybrid algorithms 205

6.4.1 Componentwise transition 206

6.4.2 Metropolis within Gibbs 211

6.4.3 Blocking 214

6.4.4 Reparametrization 216

6.5 Applications 217

6.5.1 Generalized linear mixed models 217

6.5.2 Dynamic linear models 223

6.5.3 Dynamic generalized linear models 226

6.5.4 Spatial models 231

7 Further topics in MCMC 237

7.2 Model adequacy 237

7.2.1 Estimates of the predictive likelihood 238

7.2.2 Uses of the predictive likelihood 248

7.2.3 Deviance information criterion 253

7.3 Model choice: MCMC over model and parameter spaces 257

7.3.1 Markov chain for supermodels 258

7.3.2 Markov chain with jumps 261

7.3.3 Further issues related to RJMCMC algorithms 270

7.4 Convergence acceleration 271

7.4.1 Alterations to the chain 271

7.4.2 Alterations to the equilibrium distribution 278

7.4.3 Auxiliary variables 282.

Notes:

Includes bibliographical references (pages [289]-310) and indexes.

ISBN:

1584885874

9781584885870

OCLC:

64596046

Publisher Number:

99942974057

Online:

Publisher description