Maximum a posteriori estimates in linear inverse problems with log-concave priors are proper Bayes estimators

A frequent matter of debate in Bayesian inversion is the question of which of the two principal point-estimators, the maximum a posteriori (MAP) or the conditional mean (CM) estimate, is to be preferred. As the MAP estimate corresponds to the solution given by variational regularization techniques, this is also a constant matter of debate between the two research areas. Following a theoretical argument-the Bayes cost formalism-the CM estimate is classically preferred for being the Bayes estimator for the mean squared error cost, while the MAP estimate is classically discredited for being only asymptotically the Bayes estimator for the uniform cost function. In this article we present recent theoretical and computational observations that challenge this point of view, in particular for high-dimensional sparsity-promoting Bayesian inversion. Using Bregman distances, we present new, proper convex Bayes cost functions for which the MAP estimator is the Bayes estimator. We complement this finding with results that correct further common misconceptions about MAP estimates. In total, we aim to rehabilitate MAP estimates in linear inverse problems with log-concave priors as proper Bayes estimators.