On the validity of the formal Edgeworth expansion for posterior densities

TL;DR

This paper rigorously establishes the validity of the formal Edgeworth expansion for posterior densities in Bayesian theory, resolving a long-standing open problem and providing new theoretical insights and numerical validation.

Contribution

It proves the validity of the formal Edgeworth expansion for posterior densities and introduces a lemma on posterior cumulants, advancing Bayesian asymptotic theory.

Findings

The Edgeworth expansion for posterior densities is rigorously validated.

Numerical results show the expansion performs as expected and outperforms previous approximations.

A new lemma on posterior cumulants is introduced, of independent interest.

Abstract

We consider a fundamental open problem in parametric Bayesian theory, namely the validity of the formal Edgeworth expansion of the posterior density. While the study of valid asymptotic expansions for posterior distributions constitutes a rich literature, the validity of the formal Edgeworth expansion has not been rigorously established. Several authors have claimed connections of various posterior expansions with the classical Edgeworth expansion, or have simply assumed its validity. Our main result settles this open problem. We also prove a lemma concerning the order of posterior cumulants which is of independent interest in Bayesian parametric theory. The most relevant literature is synthesized and compared to the newly-derived Edgeworth expansions. Numerical investigations illustrate that our expansion has the behavior expected of an Edgeworth expansion, and that it has better performance than the other existing expansion which was previously claimed to be of Edgeworth-type.