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

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posted on 2023-06-09, 12:02 authored by Sergios Agapiou, Martin Burger, Masoumeh DashtiMasoumeh Dashti, Tapio Helin
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.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Inverse Problems

ISSN

0266-5611

Publisher

Institute of Physics

Issue

4

Volume

34

Page range

1-37

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Analysis and Partial Differential Equations Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2018-02-12

First Open Access (FOA) Date

2019-02-20

First Compliant Deposit (FCD) Date

2018-02-12

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