# An Empirical Bayes Method for Chi-Squared Data

## TL;DR

This paper develops a Bayesian hierarchical model for chi-squared data that extends Tweedie's formula, enabling bias correction directly from data without prior assumptions, and reveals new phenomena specific to chi-squared distributions.

## Contribution

It introduces a novel hierarchical model for chi-squared data that allows Tweedie's formula to be applied, overcoming limitations of existing methods for non-exponential family distributions.

## Key findings

- Tweedie's formula can be adapted for chi-squared data.
- New phenomena in bias correction for chi-squared distributions are identified.
- The method enables bias correction without prior information.

## Abstract

In a thought-provoking paper, Efron (2011) investigated the merit and limitation of an empirical Bayes method to correct selection bias based on Tweedie's formula first reported by \cite{Robbins:1956}. The exceptional virtue of Tweedie's formula for the normal distribution lies in its representation of selection bias as a simple function of the derivative of log marginal likelihood. Since the marginal likelihood and its derivative can be estimated from the data directly without invoking prior information, bias correction can be carried out conveniently. We propose a Bayesian hierarchical model for chi-squared data such that the resulting Tweedie's formula has the same virtue as that of the normal distribution. Because the family of noncentral chi-squared distributions, the common alternative distributions for chi-squared tests, does not constitute an exponential family, our results cannot be obtained by extending existing results. Furthermore, the corresponding Tweedie's formula manifests new phenomena quite different from those of the normal distribution and suggests new ways of analyzing chi-squared data.