Adaptive Partition Weighted Approach for Estimating Marginal Posterior Density with Applications

Abstract

The computation of marginal posterior density in Bayesian analysis is essential in that it can provide complete information about parameters of interest. Furthermore, the marginal posterior density can be used for computing Bayes factors, posterior model probabilities, and diagnostic measures. The conditional marginal density estimator (CMDE) is theoretically the best for marginal density estimation but requires the closed-form expression of the conditional posterior density, which is often not available in many applications. We develop an Adaptive Partition weighTed (APT) method to realize the CMDE. This unbiased estimator requires only a single MCMC output from the joint posterior distribution and the known unnormalized posterior density. The theoretical properties and various applications of the APT estimator are examined in detail. The APT method is also extended to the estimation of conditional posterior densities. We further demonstrate the desirable features of the proposed method with two real data sets: one is from a study of dissociative identity disorder patients using an analysis of variance model with constrained inequalities; the other is from a prostate cancer study, where model selection is investigated using ordinal probit regression models with latent variables. This is a joint work with Dr. Ming-Hui Chen, Dr. Lynn Kuo, and Dr. Paul O. Lewis.