R2 priors for Grouped Variance Decomposition in High-dimensional Regression

2025-07-15

AI Summary

This paper addresses high-dimensional regression with known group structures among predictors. Method: We propose the Group-R² decomposition prior—a two-stage R² prior that explicitly allocates variance across and within groups—extending the R² prior to grouped settings for the first time. Integrating hierarchical shrinkage, Bayesian hierarchical modeling, and R² decomposition, the method employs marginal distribution and tail behavior analysis to theoretically characterize model complexity and enforce sparsity control. Contribution/Results: We establish theoretical guarantees on enhanced parameter interpretability and recovery accuracy. Simulation studies demonstrate that, under prominent group structure, the proposed prior significantly outperforms non-grouped baselines in both prediction accuracy and variable selection. The core contribution is a principled, interpretable variance-decomposition framework that rigorously delineates the effectiveness boundaries and advantageous conditions for grouped priors.

Technology Category

• Reasoning under Uncertainty: Relational Probabilistic Models
• Machine Learning: Classification and Regression
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• User Modeling, Personalization and Recommendation: Explainable and interpretable methods for personalization
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs

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.

Problem

• Extends R2 priors to structured regression with grouped predictors
• Enables interpretable control over model complexity and sparsity
• Evaluates grouping impact on predictive performance and parameter recovery

Innovation

• Hierarchical shrinkage prior for grouped predictors
• Two-stage R2 decomposition for model control
• Theoretical and simulation-based performance evaluation