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

作者：Agapiou Sergios; Burger Martin; Dashti Masoumeh; Helin Tapio

来源：Inverse Problems, 2018, 34(4): 045002.

DOI：10.1088/1361-6420/aaacac

• Bayesian inverse problems
• Besov prior
• MAP estimators

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.

• 出版日期2018-4