  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.
