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Maximum-a-posteriori (MAP) estimation is the main Bayesian estimation methodology in imaging sciences, where high dimensionality is often addressed by using Bayesian models that are log-concave and whose posterior mode can be computed efficiently by convex optimization. However, despite its success and wide adoption, MAP estimation is not theoretically well understood yet. In particular, the prevalent view in the community is that MAP estimation is not proper Bayesian estimation in the sense of Bayesian decision theory because it does not minimize a meaningful expected loss function (unlike the minimum mean squared error (MMSE) estimator that minimizes the mean squared loss). This paper addresses this theoretical gap by presenting a general decision-theoretic derivation of MAP estimation in Bayesian models that are log-concave. A main novelty is that our analysis is based on differential geometry and proceeds as follows. First, we use the underlying convex geometry of the Bayesian model to induce a Rieman...

Revisiting Maximum-A-Posteriori Estimation in Log-Concave Models