Horseshoe shrinkage methods for Bayesian fusion estimation

TL;DR

This paper introduces a Bayesian fusion estimation method using Horseshoe priors for high-dimensional signals with piecewise constant or block structures, providing theoretical guarantees and demonstrating superior performance in simulations and real-world applications.

Contribution

The paper develops a novel Horseshoe prior-based Bayesian fusion method with theoretical analysis and extends it to graph signal de-noising, showing improved estimation and structure recovery.

Findings

Consistent estimation and structure recovery demonstrated.

Theoretical posterior convergence rates established.

Superior performance shown in simulations and real data applications.

Abstract

We consider the problem of estimation and structure learning of high dimensional signals via a normal sequence model, where the underlying parameter vector is piecewise constant, or has a block structure. We develop a Bayesian fusion estimation method by using the Horseshoe prior to induce a strong shrinkage effect on successive differences in the mean parameters, simultaneously imposing sufficient prior concentration for non-zero values of the same. The proposed method thus facilitates consistent estimation and structure recovery of the signal pieces. We provide theoretical justifications of our approach by deriving posterior convergence rates and establishing selection consistency under suitable assumptions. We also extend our proposed method to signal de-noising over arbitrary graphs and develop efficient computational methods along with providing theoretical guarantees. We demonstrate the superior performance of the Horseshoe based Bayesian fusion estimation method through extensive simulations and two real-life examples on signal de-noising in biological and geophysical applications. We also demonstrate the estimation performance of our method on a real-world large network for the graph signal de-noising problem.