## Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, Second EditionWhile 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 |

289 | |

311 | |

316 | |