ON THE RELATIONSHIP BETWEEN BAYESIAN AND NON-BAYESIAN INTERVAL ESTIMATES

Let Y1, ..., Y(n) denote independent observations each distributed according to a density depending on a scalar parameter-theta. Suppose that we are interested in constructing an interval estimate for theta. One approach is to construct a confidence region with a specified coverage probability based on the likelihood ratio test statistic. Another approach is to construct a highest posterior density region with a specified posterior probability. The goal of this paper is to study the relationship between these two approaches. In particular, we derive asymptotic expansions for the posterior probability of confidence regions based on the likelihood ratio test statistic and for the coverage probability of highest posterior density regions. Conditions under which the two methods lead to identical regions, at least approximately, are also given.