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.

翻译：贝叶斯统计中，后验分布有两种常见的集中趋势度量：后验均值与最大后验估计。本文探讨了最大后验估计与后验均值之间的联系。我们推导了先验密度对的一种渐近条件，在该条件下基于一个先验的后验均值与基于另一个先验的最大后验估计一致。这一先验对存在的充分条件与统计模型在信息几何中的α-平坦性相关。我们还利用α-平行先验构造了匹配先验对。该结果阐明了广义线性回归模型中正则化与后验期望之间的有趣联系。