BAYES AND LIKELIHOOD CALCULATIONS FROM CONFIDENCE-INTERVALS

Recently there has been considerable progress on setting good approximate confidence intervals for a single parameter theta in a multi-parameter family. Here we use these frequentist results as a convenient device for making Bayes, empirical Bayes and likelihood inferences about theta. A simple formula is given that produces an approximate likelihood function L(x)dagger(theta) for theta, with all nuisance parameters eliminated, based on any system of approximate confidence intervals. The statistician can then modify L(x)dagger(theta) with Bayes or empirical Bayes information for theta, 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)dagger(theta) based on this system requires only a few times as much computation as the maximum likelihood estimate theta and its estimated standard error sigma. The formula for L(x)dagger(theta) is justified in terms of high-order adjusted likelihoods and also the Jeffreys-Welch & Peers theory of uninformative priors. Several examples are given.