  The major goal of this paper is to study the second order frequentist
properties of the marginal posterior distribution of the parametric component
in semiparametric Bayesian models, in particular, a second order semiparametric
Bernstein-von Mises (BvM) Theorem. Our first contribution is to discover an
interesting interference phenomenon between Bayesian estimation and frequentist
inferential accuracy: more accurate Bayesian estimation on the nuisance
function leads to higher frequentist inferential accuracy on the parametric
component. As the second contribution, we propose a new class of dependent
priors under which Bayesian inference procedures for the parametric component
are not only efficient but also adaptive (w.r.t. the smoothness of
nonparametric component) up to the second order frequentist validity. However,
commonly used independent priors may even fail to produce a desirable root-n
contraction rate for the parametric component in this adaptive case unless some
stringent assumption is imposed. Three important classes of semiparametric
models are examined, and extensive simulations are also provided.
