  We revisit the question of priors that achieve approximate matching of
Bayesian and frequentist predictive probabilities. Such priors may be thought
of as providing frequentist calibration of Bayesian prediction or simply as
devices for producing frequentist prediction regions. Here we analyse the
$O(n^{-1})$ term in the expansion of the coverage probability of a Bayesian
prediction region, as derived in [Ann. Statist. 28 (2000) 1414--1426]. Unlike
the situation for parametric matching, asymptotic predictive matching priors
may depend on the level $\alpha$. We investigate uniformly predictive matching
priors (UPMPs); that is, priors for which this $O(n^{-1})$ term is zero for all
$\alpha$. It was shown in [Ann. Statist. 28 (2000) 1414--1426] that, in the
case of quantile matching and a scalar parameter, if such a prior exists then
it must be Jeffreys' prior. In the present article we investigate UPMPs in the
multiparameter case and present some general results about the form, and
uniqueness or otherwise, of UPMPs for both quantile and highest predictive
density matching.
