On the Bernstein-von Mises phenomenon for nonparametric Bayes procedures. (English) Zbl 1305.62190

Summary: We continue the investigation of Bernstein-von Mises theorems for nonparametric Bayes procedures from [Ann. Stat. 41, No. 4, 1999–2028 (2013; Zbl 1285.62052)]. 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.


62G20 Asymptotic properties of nonparametric inference
62F15 Bayesian inference
62G08 Nonparametric regression and quantile regression
62G15 Nonparametric tolerance and confidence regions


Zbl 1285.62052
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