Bernstein-von Mises I: Functionals

This chapter goes beyond posterior contraction rates and considers deriving more precise results on the limiting shape of posterior distributions. We explain how to do so for certain smooth enough parameters, estimable at \(1/\sqrt {n}\) rate. After briefly reviewing existing results in parametric models, we discuss in details a more general semiparametric setting, where one wishes to estimate a finite-dimensional functional, for instance a mean or a quantile. We derive a Bernstein–von Mises theorem in this setting under some generic conditions. This result states that the posterior distribution induced on the functional of interest asymptotically resembles a specific Gaussian distribution. We then illustrate these results through a number of examples of priors, including Gaussian process priors.