- Bayesian Anal. (2017), 27 pages.
The Horseshoe+ Estimator of Ultra-Sparse Signals
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.
Bayesian Anal. (2017), 27 pages.
First available in Project Euclid: 22 September 2016
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Bhadra, Anindya; Datta, Jyotishka; Polson, Nicholas G.; Willard, Brandon. The Horseshoe+ Estimator of Ultra-Sparse Signals. Bayesian Anal., advance publication, 22 September 2016. doi: 10.1214/16-BA1028. http://projecteuclid.org/euclid.ba/1474572263
- Supplementary Material to “The Horseshoe+ Estimator of Ultra-Sparse Signals”. Contains the proofs of theorems and an additional simulation example.