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August 2020 On the validity of the formal Edgeworth expansion for posterior densities
John E. Kolassa, Todd A. Kuffner
Ann. Statist. 48(4): 1940-1958 (August 2020). DOI: 10.1214/19-AOS1871


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.


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John E. Kolassa. Todd A. Kuffner. "On the validity of the formal Edgeworth expansion for posterior densities." Ann. Statist. 48 (4) 1940 - 1958, August 2020.


Received: 1 October 2017; Revised: 1 March 2019; Published: August 2020
First available in Project Euclid: 14 August 2020

MathSciNet: MR4134781
Digital Object Identifier: 10.1214/19-AOS1871

Primary: 62E20, 62F15
Secondary: 62F99

Rights: Copyright © 2020 Institute of Mathematical Statistics


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Vol.48 • No. 4 • August 2020
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