Shrinkage with Robustness: Log-Adjusted Priors for Sparse Signals

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.

Sparse Bayesian inference on gamma-distributed observations using shape-scale inverse-gamma mixtures

“…Under Gaussian sequence or normal linear regression models, there have been proposed a variety of shrinkage priors including the most famous horseshoe (Carvalho et al, 2010) prior and related priors (e.g. Armagan et al, 2013;Bhadra, et al, 2017;Bhattacharya et al, 2015;Hamura et al, 2020;Zhang et al, 2020). This prior has an attractive shrinkage property, making it possible to strongly shrink small observations toward zero while keeping large observations unshrunk.…”