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

## TL;DR

This paper establishes that in non-parametric Bayesian inverse problems with Besov space priors, MAP estimates can be characterized as minimizers of a generalized functional, even with non-Gaussian priors, and demonstrates their consistency.

## Contribution

It proves that MAP estimates in non-parametric Bayesian inverse problems with Besov priors are characterized by a generalized Onsager--Machlup functional, extending previous results to non-Gaussian priors.

## Key findings

- MAP estimates are characterized by a generalized Onsager--Machlup functional.
- Weak and strong MAP estimates coincide in this setting.
- Weak consistency of MAP estimators is proven in the high data limit.

## 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.