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The major goal of this paper is to study the second order frequentistproperties of the marginal posterior distribution of the parametric componentin semiparametric Bayesian models, in particular, a second order semiparametricBernstein-von Mises (BvM) Theorem. Our first contribution is to discover aninteresting interference phenomenon between Bayesian estimation and frequentistinferential accuracy: more accurate Bayesian estimation on the nuisancefunction leads to higher frequentist inferential accuracy on the parametriccomponent. As the second contribution, we propose a new class of dependentpriors under which Bayesian inference procedures for the parametric componentare not only efficient but also adaptive (w.r.t. the smoothness ofnonparametric component) up to the second order frequentist validity. However,commonly used independent priors may even fail to produce a desirable root-ncontraction rate for the parametric component in this adaptive case unless somestringent assumption is imposed. Three important classes of semiparametricmodels are examined, and extensive simulations are also provided.

Semiparametric Bernstein-von Mises Theorem: Second Order Studies