On numerical implementation of α-stable priors in Bayesian inversion

Abstract

In this thesis, we introduce numerical approximations of Levy α-stable random field priors for Bayesian inversion. The α-stable processes are well-studied in stochastic process literature, and they can be potentially formulated as discretization-invariant priors. The work is also motivated by the fact that Gaussian and Cauchy distributions are both part of α-stable distributions. In Bayesian inversion, Gaussian priors are prevalent choices if the unknown should be smooth, while the Cauchy are good options for discontinuity-preserving or sparsity-promoting scenarios. One of our objectives is to construct a systematic numerical treatment of the α-stable priors to favor the coexistence of heterogeneous features, which is predominantly done through hierarchical priors or mixture models. However, the probability density functions of the α-stable distributions cannot be expressed through elementary functions in general. We address the issue by introducing a hybrid method to approximate the symmetric univariate and bivariate α-stable log probability density functions. The method is fast to evaluate, works for a continuous range of stability indices, and is accurate for Bayesian inversion.

We demonstrate the practical properties of the α-stable priors in high-dimensional Bayesian inverse problems. We employ several different α-stable field priors, including the difference priors and Bayesian neural networks. While the α-stable priors offer substantial novelties for the inversion, performing full inference with them is difficult due to their heavy-tailedness. This issue is illustrated with the help of advanced Markov chain Monte Carlo methods, which are unable to sample the posteriors with α-stable priors satisfactorily. We conclude the work by arguing that α-stable priors would significantly benefit from advanced inference methods. Additionally, the presented work offers a foundation for discretizing α-stable random field priors on unstructured meshes or with Karhunen-Loeve-type expansions.