The Horseshoe+ Estimator of Ultra-Sparse Signals

TL;DR

The paper introduces the horseshoe+ prior, an advanced Bayesian method for ultra-sparse signal detection, demonstrating faster convergence, lower estimation error, and superior performance over existing methods through theoretical analysis and simulations.

Contribution

It proposes the horseshoe+ prior, extending the horseshoe prior with improved theoretical properties and computational feasibility for ultra-sparse signals.

Findings

Faster posterior concentration than horseshoe in K-L sense

Lower mean squared error in signal estimation

Superior performance in simulations and real data analysis

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.