Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second Edition

Front Cover
CRC Press, May 10, 2006 - Mathematics - 344 pages
While there have been few theoretical contributions on the Markov Chain Monte Carlo (MCMC) methods in the past decade, current understanding and application of MCMC to the solution of inference problems has increased by leaps and bounds. 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.

Major changes from the previous edition:

More examples with discussion of computational details in chapters on Gibbs sampling and Metropolis-Hastings algorithms

Recent developments in MCMC, including reversible jump, slice sampling, bridge sampling, path sampling, multiple-try, and delayed rejection

Discussion of computation using both R and WinBUGS

Additional exercises and selected solutions within the text, with all data sets and software available for download from the Web

Sections on spatial models and model adequacy

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.

 

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Contents

Stochastic simulation
9
12 Generation of discrete random quantities
10
121 Bernoulli distribution
11
123 Geometric and negative binomial distribution
12
13 Generation of continuous random quantities
13
132 Bivariate techniques
14
133 Methods based on mixtures
17
14 Generation of random vectors and matrices
20
49 Data augmentation or substitution sampling
135
410 Exercises
136
Gibbs sampling
141
52 Definition and properties
142
53 Implementation and optimization
148
532 Scanning strategies
150
555 Using the sample
151
534 Reparametrization
152

141 Multivariate normal distribution
21
142 Wishart distribution
23
143 Multivariate Students t distribution
24
15 Resampling methods
25
152 Weighted resampling method
30
153 Adaptive rejection method
32
16 Exercises
34
Bayesian inference
41
221 Prior posterior and predictive distributions
42
222 Summarizing the information
47
23 Conjugate distributions
49
231 Conjugate distributions for the exponential family
51
232 Conjugacy and regression models
55
233 Conditional conjugacy
58
24 Hierarchical models
60
25 Dynamic models
63
251 Sequential inference
64
252 Smoothing
65
253 Extensions
67
26 Spatial models
68
27 Model comparison
72
28 Exercises
74
Approximate methods of inference
81
32 Asymptotic approximations
82
321 Normal approximations
83
322 Mode calculation
86
323 Standard Laplace approximation
88
324 Exponential form Laplace approximations
90
33 Approximations by Gaussian quadrature
93
34 Monte Carlo integration
95
35 Methods based on stochastic simulation
98
351 Bayes theorem via the rejection method
100
352 Bayes theorem via weighted resampling
101
353 Application to dynamic models
104
36 Exercises
106
Markov chains
113
42 Definition and transition probabilities
114
43 Decomposition of the state space
118
44 Stationary distributions
121
45 Limiting theorems
124
46 Reversible chains
127
47 Continuous state spaces
129
472 Stationarity and limiting results
131
48 Simulation of a Markov chain
132
535 Blocking
155
536 Sampling from the full conditional distributions
156
54 Convergence diagnostics
157
541 Rate of convergence
158
542 Informal convergence monitors
159
543 Convergence prescription
161
544 Formal convergence methods
164
55 Applications
169
552 Dynamic models
172
553 Spatial models
176
56 MCMCbased software for Bayesian modeling
178
BUGS code for Example 57
182
Appendix 5B BUGS code for Example 58
184
MetropolisHastings algorithms
191
62 Definition and properties
193
63 Special cases
198
633 Independence chains
199
634 Other forms
204
64 Hybrid algorithms
205
641 Componentwise transition
206
642 Metropolis within Gibbs
211
643 Blocking
214
644 Reparametrization
216
65 Applications
217
652 Dynamic linear models
223
653 Dynamic generalized linear models
226
654 Spatial models
231
66 Exercises
234
Further topics in MCMC
237
721 Estimates of the predictive likelihood
238
722 Uses of the predictive likelihood
248
723 Deviance information criterion
253
MCMC over model and parameter spaces
257
731 Markov chain for supermodels
258
732 Markov chain with jumps
261
733 Further issues related to RJMCMC algorithms
270
74 Convergence acceleration
271
742 Alterations to the equilibrium distribution
278
743 Auxiliary variables
282
75 Exercises
284
References
289
Author index
311
Subject index
316
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