Accurate Approximations for Posterior Moments and Marginal Densities

TLDR

Approximate marginal posterior densities resemble saddle‑point approximations for sampling distributions. The article develops approximations for posterior means and variances of positive functions of parameters and for marginal posterior densities of arbitrary parameters. The method requires maximizing slightly modified likelihoods and evaluating observed information at the maxima, assuming the likelihood times prior is unimodal. These approximations enable computation of predictive densities and are generally as accurate or better than third‑order likelihood expansions. Keywords: Bayesian inference, Laplace method, asymptotic expansions, computation of integrals.

Abstract

Abstract This article describes approximations to the posterior means and variances of positive functions of a real or vector-valued parameter, and to the marginal posterior densities of arbitrary (i.e., not necessarily positive) parameters. These approximations can also be used to compute approximate predictive densities. To apply the proposed method, one only needs to be able to maximize slightly modified likelihood functions and to evaluate the observed information at the maxima. Nevertheless, the resulting approximations are generally as accurate and in some cases more accurate than approximations based on third-order expansions of the likelihood and requiring the evaluation of third derivatives. The approximate marginal posterior densities behave very much like saddle-point approximations for sampling distributions. The principal regularity condition required is that the likelihood times prior be unimodal. Key Words: Bayesian inferenceLaplace methodAsymptotic expansionsComputation of integrals