Revisiting maximum-a-posteriori estimation in log-concave models: from differential geometry to decision theory

Maximum-a-posteriori estimation has become the main Bayesian estimation methodology in many areas of modern data science such as mathematical imaging and machine learning, where high dimensionality is addressed by using models that are log-concave and where the posterior mode can be computed very efficiently by using convex optimisation algorithms. However, despite its success and rapid adoption, maximum-a-posteriori estimation is not theoretically well understood yet. This paper presents a new decision-theoretic derivation of maximum-a-posteriori estimation in Bayesian models that are log-concave. Our analysis is based on differential geometry and proceeds as follows. First, we exploit the log-concavity of the model to induce a Riemannian geometry on the parameter space, and use differential geometry to identify the natural or canonical loss function to perform Bayesian point estimation in that space. We then show that for log-concave models the canonical loss is the Bregman divergence associated with the negative log posterior density, and that the maximum-a-posteriori estimator is the Bayesian estimator that minimises the expected loss. We also show that the posterior mean or minimum mean square error estimator is the Bayesian estimator that minimises the dual canonical loss, and establish general performance guarantees for both maximum-a-posteriori and minimum mean square error estimation. These results provide a new understanding of these estimation methodologies under log-concavity, and reveal new insights about their good empirical performance and about the roles that log-concavity plays in high dimensional inference problems.