R2 priors for Grouped Variance Decomposition in High-dimensional Regression

TL;DR

This paper introduces the Group-R2 decomposition prior, a hierarchical shrinkage method for structured high-dimensional regression that improves interpretability and model control by leveraging group information.

Contribution

It extends R2-based priors to structured settings with groups, providing theoretical insights and practical guidance for better model sparsity and prediction.

Findings

Grouping can enhance predictive accuracy in high-dimensional regression.

The prior's tail behavior supports effective shrinkage of irrelevant predictors.

Simulation results highlight when grouping improves parameter recovery.

Abstract

We introduce the Group-R2 decomposition prior, a hierarchical shrinkage prior that extends R2-based priors to structured regression settings with known groups of predictors. By decomposing the prior distribution of the coefficient of determination R2 in two stages, first across groups, then within groups, the prior enables interpretable control over model complexity and sparsity. We derive theoretical properties of the prior, including marginal distributions of coefficients, tail behavior, and connections to effective model complexity. Through simulation studies, we evaluate the conditions under which grouping improves predictive performance and parameter recovery compared to priors that do not account for groups. Our results provide practical guidance for prior specification and highlight both the strengths and limitations of incorporating grouping into R2-based shrinkage priors.