Using Log Cauchy Priors for Modeling Sparsity

• Activity Number: 424 - Priors and Model Specifications for Variable and Feature Selection
• Type: Contributed
• Date/Time: Wednesday, August 10, 2022 : 10:30 AM to 12:20 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Zihan Zhu* and Xueying Tang
• Companies: The University of Arizona and The university of Arizona
• Keywords: Bayesian methods; Shrinkage; Sparsity

Abstract

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 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 shows tail robustness. Meanwhile, the posterior contraction rate is a sharp minimax. These properties can be generalized to any priors with a tail order arbitrarily close to one. We will also demonstrate these theoretical properties through simulations.