# Adaptive posterior concentration rates for sparse high-dimensional linear regression with random design and unknown error variance

arXiv: https://arxiv.org/abs/2405.19016 · HTML: https://mlsanity.com/paper/2405.19016/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 23 of its 69 references that are in the map.
- First posted on arXiv 29 May 2024.

## Builds on (most cited first, 23 of 23)

- [A General Framework for Updating Belief Distributions](https://mlsanity.com/paper/1306.6430/) (2013)
- [Inconsistency of Bayesian Inference for Misspecified Linear Models, and a Proposal for Repairing It](https://mlsanity.com/paper/1412.3730/) (2014)
- [Concentration of tempered posteriors and of their variational approximations](https://mlsanity.com/paper/1706.09293/) (2017)
- [Robust Generalised Bayesian Inference for Intractable Likelihoods](https://mlsanity.com/paper/2104.07359/) (2021)
- [Variational Bayes for high-dimensional linear regression with sparse priors](https://mlsanity.com/paper/1904.07150/) (2019)
- [Bayesian linear regression with sparse priors](https://mlsanity.com/paper/1403.0735/) (2014)
- [General Bayesian Updating and the Loss-Likelihood Bootstrap](https://mlsanity.com/paper/1709.07616/) (2017)
- [Calibrating general posterior credible regions](https://mlsanity.com/paper/1509.00922/) (2015)
- [High-dimensional classification by sparse logistic regression](https://mlsanity.com/paper/1706.08344/) (2017)
- [Consistency of Variational Bayes Inference for Estimation and Model Selection in Mixtures](https://mlsanity.com/paper/1805.05054/) (2018)
- [Spike and slab variational Bayes for high dimensional logistic regression](https://mlsanity.com/paper/2010.11665/) (2020)
- [On the Robustness to Misspecification of α-Posteriors and Their Variational Approximations](https://mlsanity.com/paper/2104.08324/) (2021)
- [General Bayesian Loss Function Selection and the use of Improper Models](https://mlsanity.com/paper/2106.01214/) (2021)
- [Empirical Bayes posterior concentration in sparse high-dimensional linear models](https://mlsanity.com/paper/1406.7718/) (2014)
- [A reduced-rank approach to predicting multiple binary responses through machine learning](https://mlsanity.com/paper/2306.05829/) (2023)
- [High-dimensional sparse classification using exponential weighting with empirical hinge loss](https://mlsanity.com/paper/2312.12952/) (2023)
- [Adaptation of the Tuning Parameter in General Bayesian Inference with Robust Divergence](https://mlsanity.com/paper/2106.06902/) (2021)
- [Concentration properties of fractional posterior in 1-bit matrix completion](https://mlsanity.com/paper/2404.08969/) (2024)
- [Bayesian sparse multiple regression for simultaneous rank reduction and variable selection](https://mlsanity.com/paper/1612.00877/) (2016)
- [Empirical priors for prediction in sparse high-dimensional linear regression](https://mlsanity.com/paper/1903.00961/) (2019)
- [Semiparametric inference using fractional posteriors](https://mlsanity.com/paper/2301.08158/) (2023)
- [Spike-and-Slab Group Lassos for Grouped Regression and Sparse Generalized Additive Models](https://mlsanity.com/paper/1903.01979/) (2019)
- [Bayesian inference in high-dimensional linear models using an empirical correlation-adaptive prior](https://mlsanity.com/paper/1810.00739/) (2018)

## Built on by (most cited first, 1 of 1)

- [High-dimensional Bayesian Tobit regression for censored response with Horseshoe prior](https://mlsanity.com/paper/2505.08288/) (2025)

## Abstract (arXiv)

This paper investigates sparse high-dimensional linear regression, particularly examining the properties of the posterior under conditions of random design and unknown error variance. We provide consistency results for the posterior and analyze its concentration rates, demonstrating adaptiveness to the unknown sparsity level of the regression coefficient vector. Furthermore, we extend our investigation to establish concentration outcomes for parameter estimation using specific distance measures. These findings are in line with recent discoveries in frequentist studies. Additionally, by employing techniques to address model misspecification through a fractional posterior, we broaden our analysis through oracle inequalities to encompass the critical aspect of model misspecification for the regular posterior. Our novel findings are demonstrated using two different types of sparsity priors: a shrinkage prior and a spike-and-slab prior.