The Group R2D2 Shrinkage Prior for Sparse Linear Models with Grouped Covariates

TL;DR

This paper introduces the Group R2D2 prior, a Bayesian shrinkage method designed for grouped sparsity in high-dimensional linear regression, improving variable selection and prediction accuracy.

Contribution

It extends the R2D2 prior to handle grouped covariates, providing a flexible, adaptive shrinkage mechanism at both group and individual levels.

Findings

Outperforms traditional shrinkage priors in simulations

Provides theoretical properties of the proposed prior

Demonstrates improved estimation and prediction in real data

Abstract

Shrinkage priors are a popular Bayesian paradigm to handle sparsity in high-dimensional regression. Still limited, however, is a flexible class of shrinkage priors to handle grouped sparsity, where covariates exhibit some natural grouping structure. This paper proposes a novel extension of the $R^2$-induced Dirichlet Decomposition (R2D2) prior to accommodate grouped variable selection in linear regression models. The proposed method, called the Group R2D2 prior, employs a Dirichlet prior distribution on the coefficient of determination for each group, allowing for a flexible and adaptive shrinkage that operates at both group and individual variable levels. This approach improves the original R2D2 prior to handle grouped predictors, providing a balance between within-group dependence and group-level sparsity. We present several theoretical properties of this proposed prior distribution while also developing a Markov Chain Monte Carlo algorithm. Through simulation studies and real-data analysis, we demonstrate that our method outperforms traditional shrinkage priors in terms of both estimation accuracy, inference and prediction.