Highest posterior density regions with approximate frequentist validity: The role of data-dependent priors

For the general multiparameter case, we consider the problem of ensuring frequentist validity of highest posterior density regions with margin of error o(n−1)o(n−1), where nn is the sample size. The role of data-dependent priors is investigated and it is seen that the resulting probability matching condition readily allows solutions, in contrast to what happens with data-free priors. Moreover, use of data-dependent priors is seen to be helpful even for models, such as mixture models, where closed form expressions for the expected information elements do not exist.