The Annals of Statistics

Nonparametric Bernstein–von Mises theorems in Gaussian white noise

Ismaël Castillo and Richard Nickl

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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 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 nonconjugate product priors defined on general orthonormal bases of $L^{2}$ satisfying weak conditions.

Article information

Ann. Statist., Volume 41, Number 4 (2013), 1999-2028.

First available in Project Euclid: 23 October 2013

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Zentralblatt MATH identifier

Primary: 62G20: Asymptotic properties
Secondary: 62G15: Tolerance and confidence regions 62G08: Nonparametric regression

Bayesian inference plug-in property efficiency


Castillo, Ismaël; Nickl, Richard. Nonparametric Bernstein–von Mises theorems in Gaussian white noise. Ann. Statist. 41 (2013), no. 4, 1999--2028. doi:10.1214/13-AOS1133.

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