Spatial applications of Markov chain Monte Carlo for Bayesian inference

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. This thesis considers spatial applications of MCMC for Bayesian inference. Original contributions of this thesis include (1) new auxiliary variable methods for MCMC, in particular a scheme called partial decoupling for Ising models with external field. (2) a Bayesian formulation for agricultural field trials which accounts for outliers and fertility jumps. (3) a fully Bayesian analysis of a binary classification/image restoration problem.; Markov chain Monte Carlo is introduced along with standard methods for implementation. Additionally, the methods of auxiliary variables and simulated tempering are also presented. Both of these methods may lead to improved efficiency over a straight forward implementation of MCMC. Auxiliary variable methods, which include the Swendsen-Wang algorithm from statistical physics, prove to be useful in imaging and spatial classification problems.; Pairwise difference prior distributions, a simple, yet flexible class of Markov random fields, are used for spatial modeling and image analysis. Specifically, two models are proposed which are less restrictive than more standard prior distributions. These priors allow uncertainty about the spatial process and uncertainty about the appropriate model to be incorporated into the analysis.; Two main applications are considered. The first is an analysis of an agricultural variety trial where the formulation includes a spatial model for the fertility process and allows for the possibility of outliers and jumps in fertility. This induces a considerable amount of uncertainty about the nature and location of these anomalies. The second application is a classification problem from archeology. Here, a prior distribution motivated by simulated tempering is constructed that allows for a fully Bayesian analysis.