  In this paper, inference for the parametric component of a semiparametric
model based on sampling from the posterior profile distribution is thoroughly
investigated from the frequentist viewpoint. The higher-order validity of the
profile sampler obtained in Cheng and Kosorok [Ann. Statist. 36 (2008)] is
extended to semiparametric models in which the infinite dimensional nuisance
parameter may not have a root-$n$ convergence rate. This is a nontrivial
extension because it requires a delicate analysis of the entropy of the
semiparametric models involved. We find that the accuracy of inferences based
on the profile sampler improves as the convergence rate of the nuisance
parameter increases. Simulation studies are used to verify this theoretical
result. We also establish that an exact frequentist confidence interval
obtained by inverting the profile log-likelihood ratio can be estimated with
higher-order accuracy by the credible set of the same type obtained from the
posterior profile distribution. Our theory is verified for several specific
examples.
