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

**Authors:** Samhita Pal, Subhashis Ghoshal  
**arXiv:** [2410.16577](https://arxiv.org/abs/2410.16577)  
**Date:** 2024-10-21  
**Summary page:** https://www.summarizepaper.com/en/arxiv-id/2410.16577v1/

## Key points

- Authors Samhita Pal and Subhashis Ghoshal introduce a novel Bayesian approach for high-dimensional linear regression models with sparsity
- The study addresses scenarios where the number of predictors greatly exceeds the sample size, posing a challenge in traditional regression analysis
- Sparse projection-posterior distribution effectively contracts around the true parameter value at an optimal rate tailored to the sparsity level of the vector
- True sparsity structure receives substantial sparse projection-posterior probability, indicating successful variable selection
- Recentred credible ball exhibits correct asymptotic frequentist coverage
- Proposal to distribute computational burden across multiple machines for efficient processing
- Extensive simulation studies confirm proposed approach performs well for finite sample sizes
- Application of method to real-world datasets including ADNI data shows promising results compared to state-of-the-art methods

## Summary

In their paper titled "High-dimensional Grouped-regression using Bayesian Sparse Projection-posterior," authors Samhita Pal and Subhashis Ghoshal introduce a novel Bayesian approach for estimation, uncertainty quantification, and variable selection in high-dimensional linear regression models with sparsity. The study addresses scenarios where the number of predictors greatly exceeds the sample size, presenting a significant challenge in traditional regression analysis. The authors begin by employing a conjugate normal prior without considering sparsity. Their results demonstrate that this sparse projection-posterior distribution effectively contracts around the true parameter value at an optimal rate tailored to the sparsity level of the vector. Furthermore, they show that the true sparsity structure receives substantial sparse projection-posterior probability, indicating successful variable selection. Additionally, the authors establish that a suitably recentred credible ball exhibits correct asymptotic frequentist coverage. They also propose a method to distribute computational burden across multiple machines, each handling a fraction of the dataset. Through extensive simulation studies across various settings, they confirm that their proposed approach performs well for finite sample sizes. To validate their method's efficacy in real-world applications, they apply it to several datasets including ADNI data and compare its performance against state-of-the-art methods. The authors have implemented their methodology in an \texttt{R} package named \texttt{sparseProj}, utilizing it for all computations conducted in this study. Overall, Pal and Ghoshal's research offers valuable insights into Bayesian estimation techniques for high-dimensional regression models under sparsity constraints, showcasing promising results through both theoretical analysis and practical applications.