Mathematical Sciences: Asymptotic Methods for Bayesian and Likelihood Analysis

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The primary goal of this research is to provide widely applicable methodology for Bayesian analysis based on accurate, higher-order asymptotic theory. The main analytical tools to be refined and employed are based on Laplace's method for approximating integrals and the theory of weak convergence of distributions. The effect of the parameterization on the size of apporximation errors will be explored and diagnostic tools for assessing the adequacy of an approximation will be obtained. Approximations for problems with multimodal posterior densities and margianl posterior densities of implicitly defined functions will be dervied. The first order behavior of posterior distributions based on general nonparametric prior distributions on densities and hazard functions will be investigated and the possibility of extending second order approximations to nonparametric, infinite dimensional problems will be explored. Bayesian methods are aimed at improving our ability to incorporate existing (prior) information into decision and prediction strategies. The Baysian approach has remained rather philosophical until the recent computer revolution. The current thrust of excitement and activity in this area derives from the recently obtained computational capability brought by the computer revolution and the critical need brought by inundation of information relevant to any subject of interest. These scientists are leaders in the effort to dramatically improve computational capabilities for Bayesian methods. Their research will create more powerful statistical techniques for comprehensively using informaion in the decision making process.