Bayesian computation and stochastic systems
Bayesian computation and stochastic systems,
by J. Besag, P. J. Green, D. Higdon and K. Mengersen.
Statistical Science, 10, 3-41. With discussion
(41-59) and rejoinder (59-66) (1995).
Markov chain Monte Carlo (MCMC) methods have been used extensively in
statistical physics over the last forty years, in spatial statistics
for the past twenty 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
represents a fundamental breakthrough in applied Bayesian modeling.
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