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Maximum-a-posteriori (MAP) estimation is the main Bayesian estimationmethodology in imaging sciences, where high dimensionality is often addressedby using Bayesian models that are log-concave and whose posterior mode can becomputed efficiently by convex optimisation. Despite its success and wideadoption, MAP estimation is not theoretically well understood yet. Theprevalent view in the community is that MAP estimation is not proper Bayesianestimation in a decision-theoretic sense because it does not minimise ameaningful expected loss function (unlike the minimum mean squared error (MMSE)estimator that minimises the mean squared loss). This paper addresses thistheoretical gap by presenting a decision-theoretic derivation of MAP estimationin Bayesian models that are log-concave. A main novelty is that our analysis isbased on differential geometry, and proceeds as follows. First, we use theunderlying convex geometry of the Bayesian model to induce a Riemanniangeometry on the parameter space. We then use differential geometry to identifythe so-called natural or canonical loss function to perform Bayesian pointestimation in that Riemannian manifold. For log-concave models, this canonicalloss is the Bregman divergence associated with the negative log posteriordensity. We then show that the MAP estimator is the only Bayesian estimatorthat minimises the expected canonical loss, and that the posterior mean or MMSEestimator minimises the dual canonical loss. We also study the question of MAPand MSSE estimation performance in large scales and establish a universal boundon the expected canonical error as a function of dimension, offering newinsights into the good performance observed in convex problems. These resultsprovide a new understanding of MAP and MMSE estimation in log-concave settings,and of the multiple roles that convex geometry plays in imaging problems.

Revisiting maximum-a-posteriori estimation in log-concave models