The Horseshoe+ Estimator of Ultra-Sparse Signals
(2015)cite arxiv:1502.00560.Abstract
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 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.