Bayesian inversion with a-stable priors

We propose using Levy a-stable distributions to construct 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 a = 1, and a = 2, respectively. Our target is to show that these priors provide a rich class of priors for modeling rough features. The main technical issue is that the a-stable probability density functions lack closed-form expressions, and this limits their applicability. For practical purposes, we need to approximate probability density functions through numerical integration or series expansions. For Bayesian inversion, the currently available approximation methods are either too time-consuming or do not function within the range of stability and radius arguments. To address the issue, we propose a new hybrid approximation method for symmetric univariate and bivariate a-stable distributions that is both fast to evaluate and accurate enough from a practical viewpoint. In the numerical implementation of a-stable random field priors, we use the constructed approximation method. We show how the constructed priors can be used to solve specific Bayesian inverse problems, such as the deconvolution problem and the inversion of a function governed by an elliptic partial differential equation. We also demonstrate hierarchical a-stable priors in the one-dimensional deconvolution problem. For all numerical examples, we use maximum a posteriori estimation. To that end, we exploit the limited-memory BFGS and its bounded variant for the estimator.