Journal
BAYESIAN ANALYSIS
Volume 12, Issue 4, Pages 1105-1131Publisher
INT SOC BAYESIAN ANALYSIS
DOI: 10.1214/16-BA1028
Keywords
Bayesian; global-local shrinkage; horseshoe; horseshoe; normal means; sparsity
Funding
- Statistical and Applied Mathematical Sciences Institute (SAMSI)
- National Science Foundation [DMS-1613063]
- Direct For Mathematical & Physical Scien
- Division Of Mathematical Sciences [1613063] Funding Source: National Science Foundation
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We propose a new prior for ultra-sparse signal detection that we term the horseshoe+ prior. The horseshoe+ prior is a natural extension of the horseshoe prior that has achieved success in the estimation and detection of sparse signals 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 posterior mean squared error in estimating signals compared to the horseshoe and achieves the optimal Bayes risk in testing up to a constant. For one-group global-local scale mixture priors, we 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.
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