  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.
