Sub-optimality of some continuous shrinkage priors

TL;DR

This paper investigates the limitations of certain continuous shrinkage priors in high-dimensional Bayesian models, revealing that many popular priors, including the Bayesian Lasso, lack sufficient posterior concentration.

Contribution

It provides a theoretical analysis showing the sub-optimality of several widely used continuous shrinkage priors in high-dimensional contexts.

Findings

Many common shrinkage priors lack adequate posterior concentration

Bayesian Lasso and similar priors are sub-optimal in high dimensions

Highlights computational and interpretational challenges of mixture priors

Abstract

Two-component mixture priors provide a traditional way to induce sparsity in high-dimensional Bayes models. However, several aspects of such a prior, including computational complexities in high-dimensions, interpretation of exact zeros and non-sparse posterior summaries under standard loss functions, has motivated an amazing variety of continuous shrinkage priors, which can be expressed as global-local scale mixtures of Gaussians. Interestingly, we demonstrate that many commonly used shrinkage priors, including the Bayesian Lasso, do not have adequate posterior concentration in high-dimensional settings.