Maximum a posteriori and Bayes estimators are two common methods of point
estimation in Bayesian Statistics. A number of references claim that maximum a posteriori
estimators are a limiting case of Bayes estimators with 0-1 loss. In this paper, we provide
a counterexample which shows that in general this claim is false. We then correct the claim
that by providing a level set condition for posterior densities such that the result holds.
Since both estimators are defined in terms of optimization problems, the tools of
variational analysis find a natural application to Bayesian point estimation.