Estimating the Marginal Likelihood Using the Arithmetic Mean Identity

Abstract

In this paper we propose a conceptually straightforward method to estimate the marginal data density value (also called the marginal likelihood). We show that the marginal likelihood is equal to the prior mean of the conditional density of the data given the vector of parameters restricted to a certain subset of the parameter space, $A$, times the reciprocal of the posterior probability of the subset $A$. This identity motivates one to use Arithmetic Mean estimator based on simulation from the prior distribution restricted to any (but reasonable) subset of the space of parameters. By trimming this space, regions of relatively low likelihood are removed, and thereby the efficiency of the Arithmetic Mean estimator is improved. We show that the adjusted Arithmetic Mean estimator is unbiased and consistent.