Modeling sparsity with super heavy-tailed priors

Sparsity is often a desired structure for parameters in high-dimensional statistical problems. Within a Bayesian framework, sparsity is usually induced by spike-and-slab priors or global-local shrinkage priors. The latter choice is often expressed as a scale mixture of normal distributions. It marginally places a polynomial-tailed distribution on the parameter. In general, a heavier-tailed distribution has a better performance in estimating sparse parameters. We consider the log Cauchy prior and, more generally, super heavy-tailed priors in the normal mean estimation problem. This class of priors is proper while having a tail order arbitrarily close to one. The resulting posterior mean is a shrinkage estimator, and the posterior contraction rate is sharp minimax. The empirical performance of these priors is demonstrated through simulations and a real data example.