Nonasymptotic approach to Bayesian semiparametric inference

The classical semiparametric Bernstein-von Mises (BvM) results is reconsidered in a non-classical setup allowing finite samples and model misspecication. 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 target parameter and in the dimension of sieve approximation of the nuisance parameter. This helps to identify the so called critical dimension p (n) of the sieve approximation of the full parameter for which the BvM result is applicable. If the bias induced by sieve approximation is small and dimension of sieve approximation is smaller then critical dimension than the BvM result is valid. In the important i.i.d. and regression cases, we show that the condition "p (n) (2) q/n is small", where q is the dimension of the target parameter and n is the sample size, leads to the BvM result under general assumptions on the model.