Empirical Bayes Methods for Combining Likelihoods

Abstract

Suppose that several independent experiments are observed, each one yielding a likelihood L _k (θ _k ) for a real-valued parameter of interest θ _k . For example, θ _k might be the log-odds ratio for a 2 × 2 table relating to the kth population in a series of medical experiments. This article concerns the following empirical Bayes question: How can we combine all of the likelihoods L _k to get an interval estimate for any one of the θ _k 's, say θ₁? The results are presented in the form of a realistic computational scheme that allows model building and model checking in the spirit of a regression analysis. No special mathematical forms are required for the priors or the likelihoods. This scheme is designed to take advantage of recent methods that produce approximate numerical likelihoods L _k (θ _k ) even in very complicated situations, with all nuisance parameters eliminated. The empirical Bayes likelihood theory is extended to situations where the θ _k 's have a regression structure as well as an empirical Bayes relationship. Most of the discussion is presented in terms of a hierarchical Bayes model and concerns how such a model can be implemented without requiring large amounts of Bayesian input. Frequentist approaches, such as bias correction and robustness, play a central role in the methodology.

Key Words:

• ABC method
• Confidence expectation
• Generalized linear mixed models
• Hierarchical Bayes
• Meta-analysis for likelihoods
• Relevance
• Special exponential families