High-dimensional properties for empirical priors in linear regression with unknown error variance

TL;DR

This paper extends Bayesian high-dimensional linear regression methods with empirical priors to cases with unknown error variance, establishing theoretical guarantees like model selection consistency and posterior contraction rates.

Contribution

It introduces a theoretical extension of empirical priors to unknown variance scenarios, ensuring desirable Bayesian properties in high-dimensional settings.

Findings

Achieves model selection consistency under sparsity

Establishes posterior contraction rates

Proves Bernstein von-Mises theorem for the model

Abstract

We study full Bayesian procedures for high-dimensional linear regression. We adopt data-dependent empirical priors introduced in [1]. In their paper, these priors have nice posterior contraction properties and are easy to compute. Our paper extend their theoretical results to the case of unknown error variance . Under proper sparsity assumption, we achieve model selection consistency, posterior contraction rates as well as Bernstein von-Mises theorem by analyzing multivariate t-distribution.