Empirical Bayes Methods for Compound Decision Problems: Principles and Applications

Table of Links

Abstract and 1. Introduction

2. The Compound Decision Paradigm
3. Parametric Priors
4. Nonparametric Prior Estimation
5. Empirical Bayes Methods for Discrete Data
6. Empirical Bayes Methods for Panel Data
7. Conclusion

Appendix A. Tweedie’s Formula

Appendix B. Predictive Distribution Comparison

References

Abstract. Empirical Bayes methods offer valuable tools for a large class of compound decision problems. In this tutorial we describe some basic principles of the empirical Bayes paradigm stressing their frequentist interpretation. Emphasis is placed on recent developments of nonparametric maximum likelihood methods for estimating mixture models. A more extensive introductory treatment will eventually be available in Koenker and Gu (2024). The methods are illustrated with an extended application to models of heterogeneous income dynamics based on PSID data.

1. Introduction

Empirical Bayes decision theory as introduced by Robbins (1951, 1956) represented a challenge to both the Wald (1950) and Savage (1954) strands of classical decision theory. Together with the revelations of Stein (1956) on the inadmissibility of the sample mean of a multivariate Gaussian vector in dimensions greater than two, Robbins’ results showed that compound decision problems, that is, ensembles of exchangeable decision problems could be fruitfully combined to yield improved decisions for the entire ensemble. In effect, prior information could be extracted from the ensemble yielding decision rules that performed better than classical procedures that treated each problem in isolation.

Evaluating the standard Gaussian density, φ, and simplifying we obtain the alternative decision rule,