Bayesian shrinkage

TL;DR

This paper analyzes various Bayesian shrinkage priors, revealing suboptimality of common choices like the Bayesian Lasso in high dimensions, and introduces the optimal Dirichlet Laplace priors with efficient computation.

Contribution

It provides theoretical insights into the properties of Bayesian shrinkage priors and proposes the Dirichlet Laplace prior as an optimal alternative with practical computational advantages.

Findings

Most common shrinkage priors are suboptimal in high-dimensional settings.

Dirichlet Laplace priors are shown to be optimal for high-dimensional sparse problems.

Simulation results demonstrate the superior performance of Dirichlet Laplace priors.

Abstract

Penalized regression methods, such as $L_1$ regularization, are routinely used in high-dimensional applications, and there is a rich literature on optimality properties under sparsity assumptions. In the Bayesian paradigm, sparsity is routinely induced through two-component mixture priors having a probability mass at zero, but such priors encounter daunting computational problems in high dimensions. This has motivated an amazing variety of continuous shrinkage priors, which can be expressed as global-local scale mixtures of Gaussians, facilitating computation. In sharp contrast to the corresponding frequentist literature, very little is known about the properties of such priors. Focusing on a broad class of shrinkage priors, we provide precise results on prior and posterior concentration. Interestingly, we demonstrate that most commonly used shrinkage priors, including the Bayesian Lasso, are suboptimal in high-dimensional settings. A new class of Dirichlet Laplace (DL) priors are proposed, which are optimal and lead to efficient posterior computation exploiting results from normalized random measure theory. Finite sample performance of Dirichlet Laplace priors relative to alternatives is assessed in simulations.