Semiparametric Bayesian inference: non-asymptotic approach

The classical parametric and semiparametric results of an asymptotic normality of the posterior distribution (Bernstein - von Mises theorem) will be reconsidered in a non-classical setup allowing finite samples and model misspecification. In the case of a finite dimensional nuisance parameter we obtain an upper bound on the error of Gaussian approximation of the posterior distribution for the target parameter which is explicit in the dimension of the nuisance and target parameters. This helps to identify the so called critical dimension of the full parameter for which the BvM result is applicable. The results are extended to the case of infinite dimensional parameters with the nuisance parameter from a Sobolev class. The general results will be accompanied with specific examples illustrating the theory.