326,022 research outputs found

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

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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 optimisation. Despite its success and wide adoption, MAP estimation is not theoretically well understood yet. The prevalent view in the community is that MAP estimation is not proper Bayesian estimation in a decision-theoretic sense because it does not minimise a meaningful expected loss function (unlike the minimum mean squared error (MMSE) estimator that minimises the mean squared loss). This paper addresses this theoretical gap by presenting a 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 Riemannian geometry on the parameter space. We then use differential geometry to identify the so-called natural or canonical loss function to perform Bayesian point estimation in that Riemannian manifold. For log-concave models, this canonical loss is the Bregman divergence associated with the negative log posterior density. We then show that the MAP estimator is the only Bayesian estimator that minimises the expected canonical loss, and that the posterior mean or MMSE estimator minimises the dual canonical loss. We also study the question of MAP and MSSE estimation performance in large scales and establish a universal bound on the expected canonical error as a function of dimension, offering new insights into the good performance observed in convex problems. These results provide a new understanding of MAP and MMSE estimation in log-concave settings, and of the multiple roles that convex geometry plays in imaging problems.Comment: Accepted for publication in SIAM Imaging Science

    The American Commitment to Public Propoganda

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    Random set based methods have provided a rigorous Bayesian framework and have been used extensively in the last decade for point object estimation. In this paper, we emphasize that the same methodology offers an equally powerful approach to estimation of so called extended objects, i.e., objects that result in multiple detections on the sensor side. Building upon the analogy between Bayesian state estimation of a single object and random finite set estimation for multiple objects, we give a tutorial on random set methods with an emphasis on multiple extended object estimation. The capabilities are illustrated on a simple yet insightful real life example with laser range data containing several occlusions.CADICSCUA
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