If, in the mid 1980's, one had asked the average statistician about the difficulties of using Bayesian Statistics, his/her most likely answer would have been"Well, there is this problem of selecting a prior distribution and then, even if one agrees on the prior, the whole Bayesian inference is...

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...