NONPARAMETRIC BERNSTEIN-VON MISES THEOREMS IN GAUSSIAN WHITE NOISE

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 Holder 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.