Bayesian Regression Using a Prior on the Model Fit: The R2-D2 Shrinkage Prior

TL;DR

This paper introduces the R2-D2 shrinkage prior for high-dimensional linear regression, which specifies a prior on the model fit rather than directly on coefficients, leading to improved posterior contraction and empirical performance.

Contribution

The authors propose a novel shrinkage prior based on the model fit, achieving optimal tail and origin behavior, and matching near-minimax posterior contraction rates.

Findings

Outperforms previous priors in concentration and tail behavior

Achieves near-minimax posterior contraction rate

Exhibits optimal $1/x$ behavior at origin and tails

Abstract

Prior distributions for high-dimensional linear regression require specifying a joint distribution for the unobserved regression coefficients, which is inherently difficult. We instead propose a new class of shrinkage priors for linear regression via specifying a prior first on the model fit, in particular, the coefficient of determination, and then distributing through to the coefficients in a novel way. The proposed method compares favourably to previous approaches in terms of both concentration around the origin and tail behavior, which leads to improved performance both in posterior contraction and in empirical performance. The limiting behavior of the proposed prior is $1/x$, both around the origin and in the tails. This behavior is optimal in the sense that it simultaneously lies on the boundary of being an improper prior both in the tails and around the origin. None of the existing shrinkage priors obtain this behavior in both regions simultaneously. We also demonstrate that our proposed prior leads to the same near-minimax posterior contraction rate as the spike-and-slab prior.