Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems

Agapiou, Sergios, Burger, Martin, Dashti, Masoumeh and Helin, Tapio (2018) Sparsity-promoting and edge-preserving maximum a posteriori estimators in non-parametric Bayesian inverse problems. Inverse Problems, 34 (4). pp. 1-37. ISSN 0266-5611

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We consider the inverse problem of recovering an unknown functional parameter u in a separable Banach space, from a noisy observation y of its image through a known possibly non-linear ill-posed map G. The data y is finite-dimensional and the noise is Gaussian. We adopt a Bayesian approach to the problem and consider Besov space priors (see Lassas et al. 2009), 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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Analysis and Partial Differential Equations Research Group
Subjects: Q Science > QA Mathematics
Q Science > QA Mathematics > QA0273 Probabilities. Mathematical statistics
Q Science > QA Mathematics > QA0299 Analysis. Including analytical methods connected with physical problems
Depositing User: Masoumeh Dashti
Date Deposited: 12 Feb 2018 10:38
Last Modified: 18 May 2018 15:52

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