Title: On the validity of the formal Edgeworth expansion for posterior densities,
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.