The posterior distribution in a nonparametric inverse problem is shown to
contract to the true parameter at a rate that depends on the smoothness of the
parameter, and the smoothness and scale of the prior. Correct combinations of
these characteristics lead to the minimax rate. The frequentist coverage of
credible sets is shown to depend on the combination of prior and true
parameter, with smoother priors leading to zero coverage and rougher priors to
conservative coverage. In the latter case credible sets are of the correct
order of magnitude.