Bayes and likelihood calculations from confidence intervals

Abstract

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 Lx†(θ) for θ, with all nuisance parameters eliminated, based on any system of approximate confidence intervals. The statistician can then modify Lx†(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 Lx†(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 Lx†(θ) is justified in terms of high-order adjusted likelihoods and also the Jeffreys-Welch & Peers theory of uninformative priors. Several examples are given.