A Study of Expansions of Posterior Distributions

Abstract

Johnson (1970) 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) 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.

Keywords:

• Edgeworth expansion
• Marginal posterior densities
• Stein's identity

Mathematics Subject Classification:

• 41
• 62

Acknowledgments

The authors would like to thank the referees for their valuable comments on the article. The authors are partially supported by the National Science Council of Taiwan.