  We study the linear ill-posed inverse problem with noisy data in the
statistical learning setting. Approximate reconstructions from random noisy
data are sought with general regularization schemes in Hilbert scale. We
discuss the rates of convergence for the regularized solution under the prior
assumptions and a certain link condition. We express the error in terms of
certain distance functions. For regression functions with smoothness given in
terms of source conditions the error bound can then be explicitly established.
