Two Modeling Strategies for Empirical Bayes Estimation

TL;DR

This paper compares two main strategies for empirical Bayes estimation—modeling on the parameter space ($g$-modeling) and the data space ($f$-modeling)—evaluating their accuracy and practical strengths through examples.

Contribution

It provides a detailed comparison of $g$-modeling and $f$-modeling strategies, including new computational formulas for assessing their accuracy.

Findings

Both strategies have specific strengths and limitations.

Computational formulas help evaluate the frequentist accuracy of each approach.

Examples illustrate practical applications and differences of the two methods.

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.