In regular models, the reknown Bernstein von Mises theorem states that the posterior distribution of a quantity of interest say $\theta$ is asymptotically Gaussian with mean $\hat \theta$ and variance $V/n$ when the data are assumed to be distributed from a model $P_{0}$. It also states that under $P_0$, $\sqrt{n}( \hat \theta- \theta_0) $ is asymptotically Gaussian with mean zero and variance $V$. This duality between the asymptotic behaviour of the posterior distribution of $\theta$ and the frequentist distribution of $\hat \theta$ has important implications in terms of strong adequacy between the Bayesian and frequentist approaches.
In non regular models, a similar adequacy can happen, however the asymptotic distribution may not be Gaussian nor the concentration rate by $1/\sqrt{n}$. These results are well known in parametric models. In this talk we will present some developpements that have been obtained in both regular and non regular semi- parametric models, i.e. when the parameter of interest $\theta$ is finite dimensional but the model also includes an infinite or high dimensional nuisance parameter. We will describe a general theory that has been elaborated over the
|