Importance-Weighted Marginal Bayesian Posterior Density Estimation

Abstract

Abstract Markov chain sampling schemes generate dependent observations {Θi, 0 ≤ i ≤ n} from a full joint posterior distribution π(θdata). Frequently, only certain marginals of this full posterior density are of interest; thus an interesting problem is how to estimate the marginal posterior densities based on the dependent observations {Θi, 0 ≤ i ≤ n} from π(θ data). We propose a new 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. A bivariate normal model and a constrained linear multiple regression model are used to illustrate how to derive the IWMDE's for the marginal posterior densities.