Gibbs Sampling-Based Approaches in Estimating Marginal Densities

A proof was given that a Markov transition kernal from posterior distribution p(·|y) by Gibbs sampling has an invariant distribution p(·|y) . To estimate the marginal densities of multidimensional distribution, Gibbs sampler in conjunction with Monte Carlo inte gration and conditional probability formula was used. Under some conditions, the estimated marginal densities almost converge to the true marginal densities everywhere. Two examples were given to illustrate this result.