On priors providing frequentist validity of Bayesian inference for multiple parametric functions

We characterise priors which match, up to O(n(-1)), the posterior joint cumulative distribution function of multiple parametric functions with the corresponding frequentist cumulative distribution function. This work extends and unifies the work of Ghosh & Mukerjee (1993) and Datta & Ghosh (1995a) on the topic of probability-matching priors. A set of necessary and sufficient conditions is obtained for the above characterisation. Some of these conditions depend only on the parametric functions and not on the prior. Examples are given where the joint probability matching is possible and where it is not possible.