Empirical Bayes

Shape Constraints, Compound Decisions and Empirical Bayes Rules

A shape constrained maximum likelihood variant of the kernel based empirical Bayes rule proposed by Brown and Greenshtein (2009) for the classical Gaussian compound decision problem is described and some simulation comparisons are presented. The simulation evidence suggests that the shape constrained Bayes rule improves substantially on the performance of the unconstrained kernel estimate for the Bayes rule. Two variants of the generalized non-parametric maximum likelihood (Kiefer-Wolfowitz) Bayes rule recently proposed by Jiang and Zhang (2009) are also studied. Interior point methods of computing the Kiefer-Wolfowitz estimator are proposed that substantially improve upon the prevailing EM approach.



The paper is available in pdf. Computations for the paper rely on the Mosek, proprietary convex optimization software. An R package called MeddeR exists to connect R to Matlab to Mosek, and carry out the computations. Adventurous people who would like to try to reproduce this somewhat Rube Goldberg schema in their own environments are welcome to contact me for further details. Future work will rely on the much simplified structure provided by the Rmosek package. A new package called REBayes is now underdevelopment using Rmosek. Some quirks are still under active consideration, but I'll try to maintain a current development version here. Note that this package requires the Rmosek package which in turn requires Mosek to be installed.


A related paper on testing for homogeneity in mixture models (with Jiaying Gu and Stanislav Volgushev) is available from: arXiv.

Comments are, of course, always welcome.