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

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### Abstract

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 $${\mathcal G}$$. 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.

### Most cited references17

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### Inverse problems: A Bayesian perspective

(2010)
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### Well-posed stochastic extensions of ill-posed linear problems

(1970)
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### Edge-preserving and scale-dependent properties of total variation regularization

(2003)
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### Author and article information

###### Journal
2017-05-09
###### Article
1705.03286