Robbins' Rules: Understanding Separable Decision Rules in Empirical Bayes

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

2. The Compound Decision Paradigm

Suppose that we are faced with an exchangeable, i.e. permutation invariant, ensemble of related problems:

where φ denotes some familiar – usually exponential family – density, indexed by parameters, θi , that express the underlying heterogeneity of the problems. Our task is to choose a decision rule, δ : Y → A, mapping realizations Y = (Y1, · · ·, Yn) in the sample space, Y, to actions, a = δ(Y), in an action space A that minimize the compound risk,