Markov chain Monte Carlo sampling for evaluating multidimensional integrals with application to Bayesian computation

Recently, Markov chain Monte Carlo (MCMC) sampling methods have become widely used for determining properties of a posterior distribution. Alternative to the Gibbs sampler, we elaborate on the Hit-and-Run sampler and its generalization, a black-box sampling scheme, to generate a time-reversible Markov chain from a posterior distribution. The proof of convergence and its applications to Bayesian computation with constrained parameter spaces are provided and comparisons with the other MCMC samplers are made. In addition, we propose an importance weighted marginal density estimation (IWMDE) method. An IWMDE is obtained by averaging many dependent observations of the ratio of the full joint posterior densities multiplied by a weighting conditional density w. The asymptotic properties for the IWMDE and the guidelines for choosing a weighting conditional density w are also considered. The generalized version of IWMDE for estimating marginal posterior densities when the full joint posterior density contains analytically intractable normalizing constants is developed. Furthermore, we develop Monte Carlo methods based on Kullback-Leibler divergences for comparing marginal posterior density estimators. This article is a summary of the author's Ph.D. thesis and it was presented in the Savage Award session.