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Bernstein-von Mises theorems for nonparametric Bayes priors in the Gaussianwhite noise model are proved. It is demonstrated how such results justify Bayesmethods as efficient frequentist inference procedures in a variety of concretenonparametric problems. Particularly Bayesian credible sets are constructedthat have asymptotically exact $1-\alpha$ frequentist coverage level and whose$L^2$-diameter shrinks at the minimax rate of convergence (within logarithmicfactors) over H\"{o}lder balls. Other applications include general classes oflinear and nonlinear functionals and credible bands for auto-convolutions. Theassumptions cover nonconjugate product priors defined on general orthonormalbases of $L^2$ satisfying weak conditions.

Nonparametric Bernstein-von Mises theorems in Gaussian white noise