On the Bernstein–von Mises phenomenon for nonparametric Bayes procedures

We continue the investigation of Bernstein–von Mises theorems for nonparametric Bayes procedures from Ann. Statist. 41 (2013) 1999–2028. We introduce multiscale spaces on which nonparametric priors and posteriors are naturally defined, and prove Bernstein–von Mises theorems for a variety of priors in the setting of Gaussian nonparametric regression and in the i.i.d. sampling model. From these results we deduce several applications where posterior-based inference coincides with efficient frequentist procedures, including Donsker– and Kolmogorov–Smirnov theorems for the random posterior cumulative distribution functions. We also show that multiscale posterior credible bands for the regression or density function are optimal frequentist confidence bands.