On the Convergence of Monte Carlo Approximations to the Posterior Density
by Charles J. Geyer and Luke Tierney
Technical Report No. 579
School of Statistics
University of Minnesota
June 1, 1992
Research supported in part by grant DMS-9007833 from the National Science
Foundation.
Research supported in part by grant DMS-9005858 from the National Science
Foundation.
Abstract
The Monte Carlo approximation of the posterior density using a mixture of
complete data posteriors proposed by Tanner and Wong (1987) and Gelfand and
Smith (1990) converges almost surely and L^1 to the exact posterior. The
coverages of level sets of the approximate posterior (highest posterior
density regions) converge simultaneously in the Levy metric to the exact
coverages, as do the Monte Carlo approximations of coverages proposed by
Wei and Tanner (1990). Some results are given for problems in which the
complete data likelihood must be calculated by Monte Carlo.