The Horseshoe+ Estimator of Ultra-Sparse Signals

• Activity Number: 244
• Type: Contributed
• Date/Time: Monday, August 10, 2015 : 2:00 PM to 3:50 PM
• Sponsor: Section on Bayesian Statistical Science
• Author(s): Jyotishka Datta* and Anindya Bhadra and Nicholas G. Polson and Brandon Willard
• Companies: Duke University/SAMSI and Purdue University and The University of Chicago and The University of Chicago
• Keywords: Sparse Signal Recovery ; Global Local Shrinkage ; Heavy tailed prior ; Multiple Testing ; Bayes Oracle

Abstract

We propose a new prior for ultra-sparse signal detection that we term the 'horseshoe+' prior. The horseshoe+ prior sharpens the ability of the horseshoe prior that has achieved success in the sparse signal recovery problem and has been shown to possess a number of desirable theoretical properties while enjoying computational feasibility in high dimensions. The horseshoe+ prior builds upon these advantages. Our work proves that the horseshoe+ posterior concentrates at a rate faster than that of the horseshoe in the Kullback-Leibler (K-L) sense. We also establish theoretically that the proposed estimator has lower mean squared error in estimating signals compared to the horseshoe and achieves the risk of Bayes oracle in testing up to a multiplicative constant. We also develop a new technique for analyzing the marginal sparse prior densities using the class of Meijer-G functions. In simulations, the horseshoe+ estimator demonstrates superior performance in a standard design setting against competing methods, including the horseshoe and Dirichlet-Laplace estimators. We conclude with an illustration on a prostate cancer data set and by pointing out some directions for future research.