Abstract
Maximumaposteriori (MAP) estimation is the main Bayesian estimation methodology in imaging sciences, where high dimensionality is often addressed by using Bayesian models that are logconcave 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 decisiontheoretic derivation of MAP estimation in Bayesian models that are logconcave. 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 Riemannian geometry on the parameter space. We then use differential geometry to identify the socalled natural or canonical loss function to perform Bayesian point estimation in that Riemannian manifold. For logconcave models, this canonical loss coincides with the Bregman divergence associated with the negative log posterior density. Following on from this, we show that the MAP estimator is the only Bayesian estimator that minimizes the expected canonical loss, and that the posterior mean or MMSE estimator minimizes the dual canonical loss. We then study the question of MAP and MMSE estimation performance in high dimensions. Precisely, we establish a universal bound on the expected canonical error as a function of image dimension, providing new insights on the good empirical performance observed in convex problems. Together, these results provide a new understanding of MAP and MMSE estimation in logconcave settings, and of the multiple beneficial roles that convex geometry plays in imaging problems. Finally, we illustrate this new theory by analyzing the regularizationbydenoising Bayesian models, a class of stateoftheart imaging models where priors are defined implicitly through image denoising algorithms, and an image denoising model with a wavelet shrinkage prior.
Original language  English 

Pages (fromto)  650670 
Number of pages  21 
Journal  SIAM Journal on Imaging Sciences 
Volume  12 
Issue number  1 
DOIs  
Publication status  Published  28 Mar 2019 
Keywords
 Bayesian inference
 Convex optimization
 Decision theory
 Differential geometry
 Inverse problems
 Mathematical imaging
 Maximumaposteriori estimation
ASJC Scopus subject areas
 General Mathematics
 Applied Mathematics
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Marcelo A. Pereyra
 School of Mathematical & Computer Sciences  Professor
 School of Mathematical & Computer Sciences, Actuarial Mathematics & Statistics  Professor
Person: Academic (Research & Teaching)