Critical dimension in the semiparametric Bernstein–von Mises theorem

The classical parametric and semiparametric Bernstein–von Mises (BvM) results are reconsidered in a nonclassical setup allowing finite samples and model misspecification. In the parametric case and in the case of a finite-dimensional nuisance parameter, we establish an upper bound on the error of Gaussian approximation of the posterior distribution of the target parameter; the bound depends explicitly on the dimension of the full and target parameters and on the sample size. This helps to identify the so-called critical dimension $p_n$ of the full parameter for which the BvM result is applicable. In the important special i.i.d. case, we show that the condition "$p_n^3/n$ is small" is sufficient for the BvM result to be valid under general assumptions on the model. We also provide an example of a model with the phase transition effect: the statement of the BvM theorem fails when the dimension $p_n$ approaches $n^{1/3}$.