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Nonparametric Bernstein-von Mises theorems in Gaussian white noise. (English) Zbl 1285.62052
Summary: Bernstein-von Mises theorems for nonparametric Bayes priors in the Gaussian white noise model are proved. It is demonstrated how such results justify Bayes methods as efficient frequentist inference procedures in a variety of concrete nonparametric problems. Particularly, Bayesian credible sets are constructed that have an asymptotically exact \(1-\alpha\) frequentist coverage level and whose \(L^2\)-diameter shrinks at the minimax rate of convergence (within logarithmic factors) over Hölder balls. Other applications include general classes of linear and nonlinear functionals and credible bands for auto-convolutions. The assumptions cover non-conjugate product priors defined on general orthonormal bases of \(L^2\) satisfying weak conditions.

MSC:
62G20 Asymptotic properties of nonparametric inference
62G15 Nonparametric tolerance and confidence regions
62F15 Bayesian inference
62M99 Inference from stochastic processes
62G08 Nonparametric regression and quantile regression
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