Bernstein-von Mises II: Multiscale and Applications

This chapter continues the study of Bernstein–von Mises (BvM) theorems. Here we consider the question of estimating an infinite-dimensional object, that is, a nonparametric framework. We show that deriving such BvM results is possible in such a setting if considering sufficiently large functional spaces equipped with weak enough norms. We introduce two examples of such spaces: negative-index Sobolev spaces and multiscale Besov spaces. We then discuss two applications of these ideas: Donsker-type theorems for the posterior distribution induced on the cumulative distribution function and supremum-norm posterior contraction rates.