Bayes and likelihood calculations from confidence intervals

Abstract

SUMMARY Recently there has been considerable progress on setting good approximate confidence intervals for a single parameter θ in a multi-parameter family. Here we use these frequentist results as a convenient device for making Bayes, empirical Bayes and likelihood inferences about θ. A simple formula is given that produces an approximate likelihood function L x † (θ) for θ, with all nuisance parameters eliminated, based on any system of approximate confidence intervals. The statistician can then modify L x † (θ) with Bayes or empirical Bayes information for θ, without worrying about nuisance parameters. The method is developed for multiparameter exponential families, where there exists a simple and accurate system of approximate confidence intervals for any smoothly defined parameter. The approximate likelihood L x † (θ) based on this system requires only a few times as much computation as the maximum likelihood estimate θ and its estimated standard error σ. The formula for L x † (θ) is justified in terms of high-order adjusted likelihoods and also the Jeffreys-Welch & Peers theory of uninformative priors. Several examples are given.