  Maximum a posteriori and Bayes estimators are two common methods of point
estimation in Bayesian Statistics. It is commonly accepted 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.
