On the validity of the formal Edgeworth expansion for posterior\n densities

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