Nonparametric Bernstein–von Mises theorems in Gaussian white noise
Ismaël Castillo, Richard Nickl
The Annals of Statistics · 2013 · Volume 41 · Issue 4
Abstract
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.
Keywords
62G08, 62G15, 62G20, Bayesian inference, efficiency, plug-in property
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