Bayesian inversion with {\alpha}-stable priors

TL;DR

This paper introduces a new hybrid approximation method for symmetric { extalpha}-stable distributions to enable their use as priors in Bayesian inverse problems, demonstrating their effectiveness in deconvolution and PDE-based inversion tasks.

Contribution

It develops a fast, accurate hybrid approximation technique for { extalpha}-stable distributions, facilitating their practical application as priors in Bayesian inverse problems.

Findings

The hybrid approximation method is both fast and accurate.

{ extalpha}-stable priors effectively model rough features in inverse problems.

Hierarchical { extalpha}-stable priors improve deconvolution results.

Abstract

We propose to use L\'evy {\alpha}-stable distributions for constructing priors for Bayesian inverse problems. The construction is based on Markov fields with stable-distributed increments. Special cases include the Cauchy and Gaussian distributions, with stability indices {\alpha} = 1, and {\alpha} = 2, respectively. Our target is to show that these priors provide a rich class of priors for modelling rough features. The main technical issue is that the {\alpha}-stable probability density functions do not have closed-form expressions in general, and this limits their applicability. For practical purposes, we need to approximate probability density functions through numerical integration or series expansions. Current available approximation methods are either too time-consuming or do not function within the range of stability and radius arguments needed in Bayesian inversion. To address the issue, we propose a new hybrid approximation method for symmetric univariate and bivariate {\alpha}-stable distributions, which is both fast to evaluate, and accurate enough from a practical viewpoint. Then we use approximation method in the numerical implementation of {\alpha}-stable random field priors. We demonstrate the applicability of the constructed priors on selected Bayesian inverse problems which include the deconvolution problem, and the inversion of a function governed by an elliptic partial differential equation. We also demonstrate hierarchical {\alpha}-stable priors in the one-dimensional deconvolution problem. We employ maximum-a-posterior-based estimation at all the numerical examples. To that end, we exploit the limited-memory BFGS and its bounded variant for the estimator.