A Study of Expansions of Posterior Distributions

Johnson (1970 Johnson , R. ( 1970 ). Asymptotic expansions associated with posterior distributions . Ann. Math. Statist. 41 : 851 – 864 .Crossref] , Google Scholar]) obtained expansions for marginal posterior distributions through Taylor expansions. Here, the posterior expansion is expressed in terms of the likelihood and the prior together with their derivatives. Recently, Weng (2010 Weng , R. C. ( 2010 ). A Bayesian Edgeworth expansion by Stein's Identity . Bayesian Anal. 5 ( 4 ): 741 – 764 .Crossref], Web of Science ®] , Google Scholar]) used a version of Stein's identity to derive a Bayesian Edgeworth expansion, expressed by posterior moments. Since the pivots used in these two articles are the same, it is of interest to compare these two expansions.

We found that our O(t ^?1/2) term agrees with Johnson's arithmetically, but the O(t ^?1) term does not. The simulations confirmed this finding and revealed that our O(t ^?1) term gives better performance than Johnson's.