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