  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.
