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We consider the problem of estimation and structure learning of highdimensional signals via a normal sequence model, where the underlying parametervector is piecewise constant, or has a block structure. We develop a Bayesianfusion estimation method by using the Horseshoe prior to induce a strongshrinkage effect on successive differences in the mean parameters,simultaneously imposing sufficient prior concentration for non-zero values ofthe same. The proposed method thus facilitates consistent estimation andstructure recovery of the signal pieces. We provide theoretical justificationsof our approach by deriving posterior convergence rates and establishingselection consistency under suitable assumptions. We also extend our proposedmethod to signal de-noising over arbitrary graphs and develop efficientcomputational methods along with providing theoretical guarantees. Wedemonstrate the superior performance of the Horseshoe based Bayesian fusionestimation method through extensive simulations and two real-life examples onsignal de-noising in biological and geophysical applications. We alsodemonstrate the estimation performance of our method on a real-world largenetwork for the graph signal de-noising problem.

Horseshoe shrinkage methods for Bayesian fusion estimation