Two Modeling Strategies for Empirical Bayes Estimation

Empirical Bayes methods use the data from parallel experiments, for instance, observations $X_k\sim\mathcal{N}(Θ_k,1)$ for $k=1,2,\ldots,N$, to estimate the conditional distributions $Θ_k|X_k$. There are two main estimation strategies: modeling on the $θ$ 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.

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