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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.

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