# Parametric Mean-Field empirical Bayes in high-dimensional linear regression

arXiv: https://arxiv.org/abs/2601.16842 · HTML: https://mlsanity.com/paper/2601.16842/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 23 Jan 2026.

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

- [Empirical Bayes for correlated Gaussian sequence model](https://mlsanity.com/paper/2607.03596/) (2026)

## Abstract (arXiv)

In this paper, we consider the problem of parametric empirical Bayes estimation of an i.i.d. prior in high-dimensional Bayesian linear regression, with random design. We obtain the asymptotic distribution of the variational Empirical Bayes (vEB) estimator, which approximately maximizes a variational lower bound of the intractable marginal likelihood. We characterize a sharp phase transition behavior for the vEB estimator -- namely that it is information theoretically optimal (in terms of limiting variance) up to p=o(n2/3) while it suffers from a sub-optimal convergence rate in higher dimensions. In the first regime, i.e., when p=o(n2/3), we show how the estimated prior can be calibrated to enable valid coordinate-wise and delocalized inference, both under the empirical Bayes posterior and the oracle posterior. In the second regime, we propose a debiasing technique as a way to improve the performance of the vEB estimator beyond p=o(n2/3). Extensive numerical experiments corroborate our theoretical findings.