IMPORTANCE-WEIGHTED MARGINAL BAYESIAN POSTERIOR DENSITY-ESTIMATION

Markov chain sampling schemes generate dependent observations {Theta(i), 0 less than or equal to i less than or equal to n} from a full joint posterior distribution pi(theta\data). 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 {Theta(i), 0 less than or equal to i less than or equal to n} from pi(theta\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,v 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.