# Empirical Bayes estimation of normal means, accounting for uncertainty in estimated standard errors

arXiv: https://arxiv.org/abs/1901.10679 · HTML: https://mlsanity.com/paper/1901.10679/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 30 Jan 2019.

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

- [Diffusion Model from Scratch](https://mlsanity.com/paper/2412.10824/) (2024)

## Abstract (arXiv)

We consider Empirical Bayes (EB) estimation in the normal means problem, when the standard deviations of the observations are not known precisely, but estimated with error -- which is almost always the case in practical applications. In classical statistics accounting for estimated standard errors usually involves replacing a normal distribution with a t distribution. This suggests approaching this problem by replacing the normal assumption with a t assumption, leading to an "EB t-means problem". Here we show that an approach along these lines can indeed work, but only with some care. Indeed, a naive application of this idea is flawed, and can perform poorly. We suggest how this flaw can be remedied by a two-stage procedure, which first performs EB shrinkage estimation of the standard errors and then solves an EB t-means problem. We give numerical results illustrating the effectiveness of this remedy.