# Finite sample inference for empirical Bayesian methods

## TL;DR

This paper introduces a universal finite-sample inference method for empirical Bayesian models using holdout likelihood ratios, enabling valid confidence sets and hypothesis tests in complex, high-dimensional settings.

## Contribution

It proposes a new, general approach based on holdout likelihood ratios for constructing finite-sample valid confidence sets and hypothesis tests in empirical Bayesian inference.

## Key findings

- Method provides finite-sample valid inference in complex models.
- Numerical studies show effective and meaningful inferential statements.
- Real data applications demonstrate practical utility.

## Abstract

In recent years, empirical Bayesian (EB) inference has become an attractive approach for estimation in parametric models arising in a variety of real-life problems, especially in complex and high-dimensional scientific applications. However, compared to the relative abundance of available general methods for computing point estimators in the EB framework, the construction of confidence sets and hypothesis tests with good theoretical properties remains difficult and problem specific. Motivated by the universal inference framework of Wasserman et al. (2020), we propose a general and universal method, based on holdout likelihood ratios, and utilizing the hierarchical structure of the specified Bayesian model for constructing confidence sets and hypothesis tests that are finite sample valid. We illustrate our method through a range of numerical studies and real data applications, which demonstrate that the approach is able to generate useful and meaningful inferential statements in the relevant contexts.