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：贝叶斯统计学中通常采用两种后验分布集中趋势的度量方式：后验均值与最大后验（MAP）估计。本文探讨了MAP估计与后验均值之间的内在联系。我们推导出一对先验密度满足的渐近条件，使得基于其中一个先验的后验均值与基于另一个先验的MAP估计相吻合。该先验对存在的充分条件与信息几何中统计模型的$\alpha$-平坦性相关。我们还利用$\alpha$-平行先验构建了匹配先验对。我们的研究结果揭示了广义线性回归模型中正则化与后验期望之间有趣的理论联系。