Natalia Bochkina (University of Edinburgh)

The Bernstein – von Mises theorem: relaxing its assumptions and extending it to nonregular models

The Bernstein – von Mises theorem is an important result in Bayesian asymptotics, giving conditions under which the posterior distribution of a finite-dimensional parameter can be approximated by the Gaussian distribution. On one hand, this result quantifies consistency and efficiency of Bayesian procedures which
makes them “optimal” from the frequentist point of view, and, on the other hand,  it justifies Gaussian approximation of the posterior distribution commonly used, for instance, to simplify computation of the normalising constant and to estimate the accuracy of MCMC-based posterior summaries.
In this talk I will focus on the extensions of the Bernstein – von Mises theorem in three directions. Firstly, I will give a nonasymptotic formulation of the Bernstein – von Mises theorem under possible model misspecification. It allows quantification of the effect of model misspecification and of the sample size on the posterior distribution, in particular, when constructing the approximate credible intervals. Secondly, I will state the approximation of the posterior distribution for so called nonregular distributions, where the true value of the parameter lies on the boundary of the parameter space that occurs, for example, in image analysis and in econometrics. In these cases, the approximation of the posterior distribution is no longer Gaussian, and the convergence is faster than for the regular models. And finally, this result will also be stated for non-identifiable nonregular models, e.g. ill-posed inverse problems, where Bayesian regularisation is essential.

This work is joint with Vladimir Spokoiny (WIAS, Germany) and Peter Green (University of Bristol, UK).