Fast Gibbs sampling for high-dimensional Bayesian inversion

TL;DR

This paper introduces a fast, generalized Gibbs sampling method for high-dimensional Bayesian inverse problems, enabling efficient posterior sampling with complex priors like total variation, demonstrated on CT data.

Contribution

It extends existing Gibbs samplers to a broad class of priors, including TV, using a novel slice sampling approach for efficient high-dimensional Bayesian inference.

Findings

Enables sample-based Bayesian inference with complex priors in high dimensions.

Achieves efficient sampling for CT inversion with total variation prior.

Demonstrates practical performance through computed examples.

Abstract

Solving ill-posed inverse problems by Bayesian inference has recently attracted considerable attention. Compared to deterministic approaches, the probabilistic representation of the solution by the posterior distribution can be exploited to explore and quantify its uncertainties. In applications where the inverse solution is subject to further analysis procedures, this can be a significant advantage. Alongside theoretical progress, various new computational techniques allow to sample very high dimensional posterior distributions: In [Lucka2012], a Markov chain Monte Carlo (MCMC) posterior sampler was developed for linear inverse problems with $\ell_1$-type priors. In this article, we extend this single component Gibbs-type sampler to a wide range of priors used in Bayesian inversion, such as general $\ell_p^q$ priors with additional hard constraints. Besides a fast computation of the conditional, single component densities in an explicit, parameterized form, a fast, robust and exact sampling from these one-dimensional densities is key to obtain an efficient algorithm. We demonstrate that a generalization of slice sampling can utilize their specific structure for this task and illustrate the performance of the resulting slice-within-Gibbs samplers by different computed examples. These new samplers allow us to perform sample-based Bayesian inference in high-dimensional scenarios with certain priors for the first time, including the inversion of computed tomography (CT) data with the popular isotropic total variation (TV) prior.