Horseshoe priors for edge-preserving linear Bayesian inversion

In many large-scale inverse problems, such as computed tomography and image deblurring, characterization of sharp edges in the solution is desired. Within the Bayesian approach to inverse problems, edge-preservation is often achieved using Markov random field priors based on heavy-tailed distributions. Another strategy, popular in statistics, is the application of hierarchical shrinkage priors. An advantage of this formulation lies in expressing the prior as a conditionally Gaussian distribution depending of global and local hyperparameters which are endowed with heavy-tailed hyperpriors. In this work, we revisit the shrinkage horseshoe prior and introduce its formulation for edge-preserving settings. We discuss a sampling framework based on the Gibbs sampler to solve the resulting hierarchical formulation of the Bayesian inverse problem. In particular, one of the conditional distributions is high-dimensional Gaussian, and the rest are derived in closed form by using a scale mixture representation of the heavy-tailed hyperpriors. Applications from imaging science show that our computational procedure is able to compute sharp edge-preserving posterior point estimates with reduced uncertainty.