On the Frequentist Properties of Bayesian Nonparametric Methods

In this paper, I review the main results on the asymptotic properties of the posterior distribution in nonparametric or high-dimensional models. In particular, I explain how posterior concentration rates can be derived and what we learn from such analysis in terms of the impact of the prior distribution on high-dimensional models. These results concern fully Bayes and empirical Bayes procedures. I also describe some of the results that have been obtained recently in semiparametric models, focusing mainly on the Bernstein-von Mises property. Although these results are theoretical in nature, they shed light on some subtle behaviors of the prior models and sharpen our understanding of the family of functionals that can be well estimated for a given prior model.