Critical Dimension in the Semiparametric Bernstein von Mises Theorem
Maxim Panov, Vladimir Spokoiny
Proceedings of the Steklov Institute of Mathematics, 2014, Vol. 287, pp. 232–255
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 pn of the full parameter for which the BvM result is applicable. In the important special i.i.d. case, we show that the condition “p3 n/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 pn approaches n1/3.