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**Bayesian computation and stochastic systems. With comments and reply.**
*(English)*
Zbl 0955.62552

Summary: Markov chain Monte Carlo (MCMC) methods have been used extensively in statistical physics over the last 40 years, in spatial statistics for the past 20 and in Bayesian image analysis over the last decade. In the last five years, MCMC has been introduced into significance testing, general Bayesian inference and maximum likelihood estimation. This paper presents basic methodology of MCMC, emphasizing the Bayesian paradigm, conditional probability and the intimate relationship with Markov random fields in spatial statistics. Hastings algorithms are discussed, including Gibbs, Metropolis and some other variations. Pairwise difference priors are described and are used subsequently in three Bayesian applications, in each of which there is a pronounced spatial or temporal aspect to the modeling. The examples involve logistic regression in the presence of unobserved covariates and ordinal factors; the analysis of agricultural field experiments, with adjustment for fertility gradients; and processing of low-resolution medical images obtained by a gamma camera. Additional methodological issues arise in each of these applications and in the Appendices. The paper lays particular emphasis on the calculation of posterior probabilities and concurs with others in its view that MCMC facilitates a fundamental breakthrough in applied Bayesian modeling.

Comments: Arnoldo Frigessi (41–43), Alan E. Gelfand, Bradley P. Carlin (43–46), Charles J. Geyer (46–48), G. O. Roberts, S. K. Sahu, W. R. Gilks (49–51), Wing Hung Wong (52–53), Bin Yu (54–58), Julian Besag, Peter Green, David Higdon, Kerrie Mengersen (58–66).

Comments: Arnoldo Frigessi (41–43), Alan E. Gelfand, Bradley P. Carlin (43–46), Charles J. Geyer (46–48), G. O. Roberts, S. K. Sahu, W. R. Gilks (49–51), Wing Hung Wong (52–53), Bin Yu (54–58), Julian Besag, Peter Green, David Higdon, Kerrie Mengersen (58–66).

### MSC:

62F15 | Bayesian inference |

60J10 | Markov chains (discrete-time Markov processes on discrete state spaces) |

62J12 | Generalized linear models (logistic models) |

62K10 | Statistical block designs |

62M30 | Inference from spatial processes |

65C05 | Monte Carlo methods |

68U10 | Computing methodologies for image processing |