Bayesian Hierarchical Shrinkage Prior for Sparse Signals

• Activity Number: 570
• Type: Contributed
• Date/Time: Wednesday, August 3, 2011 : 2:00 PM to 3:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Yingbo Li*+ and Merlise Clyde
• Companies: Duke University and Duke University
• Keywords: Bayesian model averaging ; variable selection ; shrinkage ; sparsity

Abstract

In this paper, we propose a new fully Bayes approach for estimating sparse signals from background Gaussian white noise. We construct a hierarchical shrinkage prior as mixtures of Cauchy densities, which in the limit leads to a Levy random field prior. The prior can also be considered as a mixture of a point mass at zero and a heavy-tailed density which permits it to adapt to different sparsity structures. By studying the tail property of its induced marginal likelihood, we prove it has bounded influence. This hierarchical shrinkage prior shrinks small values directly towards zero while keeping large signals almost unshrunk. The infinite divisibility of our prior leads to coherent prior specifications as the number of predictors increases. Based on simulation studies, under certain circumstances, our prior can achieve higher accuracy in terms of sum of squared error, comparing with some existing Bayesian model selection approaches such as Johnstone and Silverman's Empirical Bayes estimates.