Estimating Ratios of Normalizing Constants

Abstract

A computational problem arising frequently in Bayesian inference is the computation of normalizing constants for posterior densities from which we can sample. Typically, we are interested in the ratios of such normalizing constants. For example, a Bayes factor is defined as the ratio of posterior odds versus prior odds, where posterior odds is simply a ratio of the normalizing constants of two posterior densities. Mathematically, this problem can be formulated as follows. Let π_l (θ), l = 1,2, be two densities, each of which is known up to a normalizing constant: