Heavy Tailed Horseshoe Priors

Abstract: Locally adaptive shrinkage in the Bayesian framework is achieved through the use of local-global prior distributions that model both the global level of sparsity as well as individual shrinkage parameters for mean structure parameters. The most popular of these models is the Horseshoe prior and its variants due to their spike and slab behavior involving an asymptote at the origin and heavy tails. In this article, we present an alternative Horseshoe prior that exhibits both a sharper asymptote at the origin as well as heavier tails, which we call the Heavy-tailed Horseshoe prior. We prove that mixing on the shape parameters provides improved spike and slab behavior as well as better reconstruction properties than other Horseshoe variants. A simulation study is provided to show the advantage of the heavy-tailed Horseshoe in terms of absolute error to both the truth mean structure as well as the oracle.