Bayesian modeling of highly structured systems using Markov chain Monte Carl

The use of Bayesian modeling has bloomed in recent years, especially spatial modeling with the use of Markov chain Monte Carlo as a model fitting tool. This thesis deals with Bayesian modeling, prior distributions and MCMC as a method of model fitting, and follows the work through to application.;Original contributions of this thesis include: (1) Development of a formulation to disaggregate collapsed margins in events and exposures analyses. External information on the breakdown of the exposures over a collapsed margin is used to make inference on effects of the collapsed margin. (2) Demonstration of the possibility of improper posterior distributions under the disaggregation formulation and the conditions under which they might occur. (3) Discovery and demonstration of a consequence of deterioration of the Poisson approximation to the binomial distribution to effect estimation. (4) Further investigation of some properties of heavy blocking in Gibbs samplers. (5) Reworking of a pattern theoretic method of forming Markov random fields for use as prior distributions in Bayesian image analysis, correction of an old formulation on the rectangular lattice and demonstration of a new one on the hexagonal that illustrates the potential of MRFs to achieve patterns of high complexity in image prior distributions. (6) Application of another method of constructing HSMRFs to remote sensing in forestry as part of the solution to a hard and ongoing problem. (7) Application of Bayesian spatial epidemiology and disaggregation methods to prostate cancer among non-whites in the continental U.S., giving more detailed exploration of spatial effects than previously and allowing precise quantification of several effects. (8) An application of Bayesian modeling to the problem of determination of the fundamental constants of physics.;Markov chain Monte Carlo is briefly introduced as background to the multivariate Gibbs updating methods and because the models in application chapters are fit with MCMC. Markov random fields are also briefly introduced, leading to their use in the development of image prior distributions and their application to Bayesian spatial epidemiology.;Two main applications are considered in the thesis, although work mentioned has been applied elsewhere. One application is the spatial analysis of prostate cancer in non-whites in the United States, where spatial models using disaggregation methods are employed in the search for risk factors for prostate cancer. The other main application is a dataset from physics that contains many discrepant values and falls naturally into the Bayesian framework.