Two Modeling Strategies for Empirical Bayes Estimation

Abstract

Empirical Bayes methods use the data from parallel experiments, for instance, observations $X_{k}\sim\mathcal{N}(\Theta_{k},1)$ for $k=1,2,\ldots,N$, to estimate the conditional distributions $\Theta_{k}|X_{k}$. There are two main estimation strategies: modeling on the $\theta$ space, called “$g$-modeling” here, and modeling on the $x$ space, called “$f$-modeling.” The two approaches are described and compared. A series of computational formulas are developed to assess their frequentist accuracy. Several examples, both contrived and genuine, show the strengths and limitations of the two strategies.