Abstract : The Bayesian approach is a possible way to build estimators in statistical models. It consists in attributing a probability measure -the prior- to the unknown parameters of the model. The estimator is then the posterior distribution, which is a conditional distribution given the information contained in the data.
The Bernstein-von Mises theorem in parametric models states that under mild regularity conditions, the posterior distribution for the finite-dimensional model parameter is asymptotically Gaussian with `optimal' centering and variance.
In this talk I will discuss recent advances in the understanding of posterior distributions in nonparametric models, that is when the unknown parameter is infinite-dimensional, focusing on a concept of nonparametric Bernstein-von Mises theorem.