The Horseshoe+ Estimator of Ultra-Sparse Signals

TLDR

The horseshoe prior has proven effective for sparse signal estimation, and the horseshoe+ prior extends these advantages to ultra‑sparse settings. The authors introduce the horseshoe+ prior to detect ultra‑sparse signals. They develop a Meijer‑G function–based technique to analyze marginal sparse prior densities for one‑group global–local scale mixture priors. The horseshoe+ posterior converges faster than the horseshoe in Kullback–Leibler distance, achieves lower posterior mean‑squared error and optimal Bayes risk, and in simulations outperforms competing methods such as the horseshoe and Dirichlet–Laplace, with an illustration.

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 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.