  A demanding challenge in Bayesian inversion is to efficiently characterize
the posterior distribution. This task is problematic especially in
high-dimensional non-Gaussian problems, where the structure of the posterior
can be very chaotic and difficult to analyse. Current inverse problem
literature often approaches the problem by considering suitable point
estimators for the task. Typically the choice is made between the maximum a
posteriori (MAP) or the conditional mean (CM) estimate.
  The benefits of either choice are not well-understood from the perspective of
infinite-dimensional theory. Most importantly, there exists no general scheme
regarding how to connect the topological description of a MAP estimate to a
variational problem. The results by Dashti et. al. (2013) resolve this issue
for non-linear inverse problems in Gaussian framework. In this work we improve
the current understanding by introducing a novel concept called the weak MAP
(wMAP) estimate. We show that any MAP estimate in the sense of Dashti et. al.
(2013) is a wMAP estimate and, moreover, how in general infinite-dimensional
non-Gaussian problems the wMAP estimate connects to a variational formulation.
Such a formulation yields many properties of the estimate that were earlier
impossible to study.
  In a recent work by Burger and Lucka (2014) the MAP estimator was studied in
the context of Bayes cost method. Using Bregman distances, proper convex Bayes
cost functions were introduced for which the MAP estimator is the Bayes
estimator. Here, we generalize these results to the infinite-dimensional
setting. Moreover, we discuss the implications of our results for some examples
of prior models such as the Besov prior and hierarchical prior.
