Modeling sparsity using log-Cauchy 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 heavy-tailed prior with significant probability mass around zero is preferred in estimating sparse parameters. In this talk, we consider a general class of priors, with the log Cauchy priors as a special case, 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. We also demonstrate the performance of this class of priors on simulated and real datasets.