A Bayesian Edgeworth expansion by Stein's identity

Abstract

The Edgeworth expansion is a series that approximates a probability distribution in terms of its cumulants. One can derive it by first expanding the probability distribution in Hermite orthogonal functions and then collecting terms in powers of the sample size. This paper derives an expansion for posterior distributions which possesses these features of an Edgeworth series. The techniques used are a version of Stein's Identity and properties of Hermite polynomials. Two examples are provided to illustrate the accuracy of our series.