Bayesian High-dimensional Grouped-regression using Sparse Projection-posterior

TL;DR

This paper introduces a Bayesian high-dimensional grouped regression method using sparse projection posteriors, enabling effective group selection and estimation with proven optimal contraction rates and credible set coverage, validated through simulations and real data.

Contribution

The paper develops a novel Bayesian sparse projection approach with multiple projection maps for high-dimensional grouped regression, including a debiased version for credible set coverage, with theoretical guarantees and practical applications.

Findings

Achieves optimal posterior contraction rates for estimation and prediction.

Ensures model selection consistency in high-dimensional settings.

Demonstrates robustness through extensive simulations and real data analysis.

Abstract

We present a novel Bayesian approach for high-dimensional grouped regression under sparsity. We leverage a sparse projection method that uses a sparsity-inducing map to derive an induced posterior on a lower-dimensional parameter space. Our method introduces three distinct projection maps based on popular penalty functions: the Group LASSO Projection Posterior, Group SCAD Projection Posterior, and Adaptive Group LASSO Projection Posterior. Each projection map is constructed to immerse dense posterior samples into a structured, sparse space, allowing for effective group selection and estimation in high-dimensional settings. We derive optimal posterior contraction rates for estimation and prediction, proving that the methods are model selection consistent. Additionally, we propose a Debiased Group LASSO Projection Map, which ensures exact coverage of credible sets. Our methodology is particularly suited for applications in nonparametric additive models, where we apply it with B-spline expansions to capture complex relationships between covariates and response. Extensive simulations validate our theoretical findings, demonstrating the robustness of our approach across different settings. Finally, we illustrate the practical utility of our method with an application to brain MRI volume data from the Alzheimer's Disease Neuroimaging Initiative (ADNI), where our model identifies key brain regions associated with Alzheimer's progression.