Mathematical Sciences: Strategies for Bayesian Data Analysiswith Application to Quantal Bioassay and Geographic Disease Occurrence Models

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Recently there have been major advances in the development of monte carlo strategies to obtain samples from high dimensional nonnormalized joint densities. These techniques have proved especially valuable in fitting structured random effects models. In such settings, Analytic examination of the posterior distribution is infeasible but sampling enables arbitrarily accurate estimation of any features of the posterior with the promise of fitting an enormous range of previously inaccessible models attention must now turn to the development of tools for bayesian data analysis. Consistent with the informality of the art of model development we propose to develop informal tools with an eda flavor to mesh naturally with the sampling based approaches used to fit the model. Application will be in two Biometric areas, quantal bioassay and geographic disease mapping. CONTEMPORARY STATISTICAL WORK IS DRIVEN BY SERIOUS Applications. Realistic Models in such applications will necessarily be complex incorporating many unknowns. Indeed, there may be several plausible models with collected data, the question arises of how to fit such models with several choices of models we might ask if any of them are good and which one we prefer. The role of the data analyst is to provide answers to these questions. The proposed research is intended to help the data analyst by providing a unifying strategy for carrying out the fitting and an associated set of tools for studying adequacy of and choice of models at the heart of the entire approach is the most elementary of statistical ideas - drawing samples to learn about model unknowns. We propose to utilize this approach in two biometric areas of application. One considers geographic patterns of disease occurrence. The other involves the change in toxicity to a population due to change in exposure to a toxin.