Maximum-A-Posteriori Estimates in Linear Inverse Problems with Log-concave Priors are Proper Bayes Estimators

摘要

A frequent matter of debate in Bayesian inversion is the question, which ofthe two principle point-estimators, the maximum-a-posteriori (MAP) or theconditional mean (CM) estimate is to be preferred. As the MAP estimatecorresponds 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 estimateis classically preferred for being the Bayes estimator for the mean squarederror cost while the MAP estimate is classically discredited for being onlyasymptotically the Bayes estimator for the uniform cost function. In thisarticle we present recent theoretical and computational observations thatchallenge this point of view, in particular for high-dimensionalsparsity-promoting Bayesian inversion. Using Bregman distances, we present new,proper convex Bayes cost functions for which the MAP estimator is the Bayesestimator. We complement this finding by results that correct further commonmisconceptions about MAP estimates. In total, we aim to rehabilitate MAPestimates in linear inverse problems with log-concave priors as proper Bayesestimators.