Shrinkage with Robustness: Log-Adjusted Priors for Sparse Signals

TL;DR

This paper introduces log-adjusted shrinkage priors that enhance tail robustness and shrinkage in sparse signal analysis, supported by theoretical properties and practical Gibbs sampling methods.

Contribution

It proposes a novel class of priors with heavier tails and improved robustness, extending existing beta priors with multiple log-terms for better sparse signal modeling.

Findings

Superior tail robustness demonstrated in simulations

Effective Gibbs sampling enabled by latent variable representation

Improved posterior mean squared errors in tail regions

Abstract

We introduce a new class of distributions named log-adjusted shrinkage priors for the analysis of sparse signals, which extends the three parameter beta priors by multiplying an additional log-term to their densities. The proposed prior has density tails that are heavier than even those of the Cauchy distribution and realizes the tail-robustness of the Bayes estimator, while keeping the strong shrinkage effect on noises. We verify this property via the improved posterior mean squared errors in the tail. An integral representation with latent variables for the new density is available and enables fast and simple Gibbs samplers for the full posterior analysis. Our log-adjusted prior is significantly different from existing shrinkage priors with logarithms for allowing its further generalization by multiple log-terms in the density. The performance of the proposed priors is investigated through simulation studies and data analysis.