Matching prior pairs connecting Maximum A Posteriori estimation and posterior expectation

Bayesian statistics has two common measures of central tendency of a posterior distribution: posterior means and Maximum A Posteriori (MAP) estimates. In this paper, we discuss a connection between MAP estimates and posterior means. We derive an asymptotic condition for a pair of prior densities under which the posterior mean based on one prior coincides with the MAP estimate based on the other prior. A sufficient condition for the existence of this prior pair relates to $\alpha$-flatness of the statistical model in information geometry. We also construct a matching prior pair using $\alpha$-parallel priors. Our result elucidates an interesting connection between regularization in generalized linear regression models and posterior expectation.