Burger M (2018)
Publication Status: Published
Publication Type: Journal article
Publication year: 2018
Publisher: IOP PUBLISHING LTD
Book Volume: 34
Journal Issue: 4
URI: https://arxiv.org/pdf/1705.03286.pdf
We consider the inverse problem of recovering an unknown functional parameter u in a separable Banach space, from a noisy observation vector y of its image through a known possibly non-linear map G. We adopt a Bayesian approach to the problem and consider Besov space priors (see Lassas et al (2009 Inverse Problems Imaging 3 87-122)), which are well-known for their edge-preserving and sparsity-promoting properties and have recently attracted wide attention especially in the medical imaging community.Our key result is to show that in this non-parametric setup the maximum a posteriori (MAP) estimates are characterized by the minimizers of a generalized Onsager-Machlup functional of the posterior. This is done independently for the so-called weak and strong MAP estimates, which as we show coincide in our context. In addition, we prove a form of weak consistency for the MAP estimators in the infinitely informative data limit. Our results are remarkable for two reasons: first, the prior distribution is non-Gaussian and does not meet the smoothness conditions required in previous research on non-parametric MAP estimates. Second, the result analytically justifies existing uses of the MAP estimate in finite but high dimensional discretizations of Bayesian inverse problems with the considered Besov priors.
APA:
Burger, M. (2018). Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems. Inverse Problems, 34(4). https://dx.doi.org/10.1088/1361-6420/aaacac
MLA:
Burger, Martin. "Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems." Inverse Problems 34.4 (2018).
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