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

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.

36 pages, some typos corrected, acknowledgements added

References in corpus (4)

• Tomographic inversion using ${\ell }_1$-norm regularization of wavelet coefficients
• Discretization-invariant Bayesian inversion and Besov space priors
• Well-posed Bayesian inverse problems and heavy-tailed stable quasi-Banach space priors
• Maximum a posteriori probability estimates in infinite-dimensional Bayesian inverse problems

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• Equivalence of weak and strong modes of measures on topological vector spaces
• Γ-convergence of Onsager-Machlup functionals. Part II: Infinite product measures on Banach spaces
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• Adaptive inference over Besov spaces in the white noise model using $p$-exponential priors
• Are minimizers of the Onsager-Machlup functional strong posterior modes?
• MAP estimators for nonparametric Bayesian inverse problems in Banach spaces
• Strong maximum a posteriori estimation in Banach spaces with Gaussian priors