Probability matching property of adjusted likelihoods

For models characterized by a scalar parameter, it is well known that Jeffrey's prior ensures approximate frequentist validity of posterior quantiles. We examine how far this result remains robust in the presence of nuisance parameters, when the interest parameter θ1 is scalar, a prior on θ1 alone is considered, and the analysis is based on an adjusted version of the profile likelihood, rather than the true likelihood. This provides justification, from a Bayesian viewpoint, for some popular adjustments in term of their ability to neutralize unknown nuisance parameters. The dual problem of identifying adjustments that make a given prior on θ1 probability matching in the above sense is also addressed.