Two Simulation Based Asymptotic Methods for Computing Marginal Probability

Abstract

The computation of marginal probability has been a common practice in Bayesian inference including the construction of posterior probability and evaluation of evidence provided by the data in testing hypotheses. It is possible, however, that one may encounter difficulties in computing the normalizing constant. In this paper, we propose two estimates, a nonparametric Candidate's estimate and a Laplace type approximation, to estimate the marginal probability. The first Candidate's estimate was proposed before by other statisticians but here we compute its value via Markov chain Monte Carlo method. We derive the asymptotic behavior of the Candidate's estimate and also the best point, which may not be the posterior mode, for evaluating the estimate. When the shape of the posterior distribution is highly skewed and has mode close to the boundary, the proposed estimate still performs well. The second Laplace type estimate also utilizes the simulated sample from Markov chain Monte Carlo method. We derive an optimal choice for the volume around the mode to correct for the usual Laplace's method which depends heavily on the accuracy of the estimated mode. The optimal choice is shown to be invariant under linear transformation of the posterior sample, which in turn implies an easy implementation of this approach. We will illustrate these two methods and compare with other Laplace types estimates, including Laplace-Metropolis estimate, in a simulation study and real data. (Based on joint work with Ching-Wei Chang, Division of Biostatistics, Institute of Epidemiology, National Taiwan University and Su-Yun Huang, Institute of Statistical Science, Academia Sinica)