On some problems of Bayesian region construction with guaranteed coverages

The general problem of constructing regions that have a guaranteed coverage probability for an arbitrary parameter of interest $\psi \in \Psi$ is considered. The regions developed are Bayesian in nature and the coverage probabilities can be considered as Bayesian confidences with respect to the model obtained by integrating out the nuisance parameters using the conditional prior given $\psi .$ Both the prior coverage probability and the prior probability of covering a false value (the accuracy) can be controlled by setting the sample size. These coverage probabilities are considered as a priori figures of merit concerning the reliability of a study while the inferences quoted are Bayesian. Several problems are considered where obtaining confidence regions with desirable properties have proven difficult to obtain. For example, it is shown that the approach discussed never leads to improper regions which has proven to be an issue for some confidence regions.