Efron, Bradley Empirical Bayes methods for combining likelihoods. (With discussion). (English) Zbl 0868.62018 J. Am. Stat. Assoc. 91, No. 434, 538-565 (1996). Summary: Suppose that several independent experiments are observed, each one yielding a likelihood \(L_k(\theta_k)\) for a real-valued parameter of interest \(\theta_k\). For example, \(\theta_k\) might be the log-odds ratio for a \(2\times 2\) table relating to the \(k\)th 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 \(\theta_k\)’s, say \(\theta_1\)? 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 \(L_k(\theta_k)\) even in very complicated situations, with all nuisance parameters eliminated. The empirical Bayes likelihood theory is extended to situations where the \(\theta_k\)’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. Cited in 22 Documents MSC: 62C12 Empirical decision procedures; empirical Bayes procedures 62F25 Parametric tolerance and confidence regions 62A01 Foundations and philosophical topics in statistics 62F15 Bayesian inference Keywords:ABC method; confidence expectation; generalized linear mixed models; meta-analysis for likelihoods; relevance; special exponential families; log-odds ratio; computational scheme; model building; model checking; nuisance parameters; empirical Bayes likelihood theory; hierarchical Bayes model; bias correction; robustness PDFBibTeX XMLCite \textit{B. Efron}, J. Am. Stat. Assoc. 91, No. 434, 538--565 (1996; Zbl 0868.62018) Full Text: DOI