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In many large-scale inverse problems, such as computed tomography and imagedeblurring, characterization of sharp edges in the solution is desired. Withinthe Bayesian approach to inverse problems, edge-preservation is often achievedusing Markov random field priors based on heavy-tailed distributions. Anotherstrategy, popular in statistics, is the application of hierarchical shrinkagepriors. An advantage of this formulation lies in expressing the prior as aconditionally Gaussian distribution depending of global and localhyperparameters which are endowed with heavy-tailed hyperpriors. In this work,we revisit the shrinkage horseshoe prior and introduce its formulation foredge-preserving settings. We discuss a sampling framework based on the Gibbssampler to solve the resulting hierarchical formulation of the Bayesian inverseproblem. In particular, one of the conditional distributions ishigh-dimensional Gaussian, and the rest are derived in closed form by using ascale mixture representation of the heavy-tailed hyperpriors. Applications fromimaging science show that our computational procedure is able to compute sharpedge-preserving posterior point estimates with reduced uncertainty.

Horseshoe priors for edge-preserving linear Bayesian inversion