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

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 favorably 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. Supplementary materials for this article are available online.

KEYWORDS:

• Beta-prime distribution
• Coefficient of determination
• Global-local shrinkage
• High-dimensional regression

Supplementary Materials

The supplementary materials contain an appendix with the technical details and all proofs, and also includes additional details on tuning of the hyperparameters in the approach.