  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 H\"{o}lder 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.
