Regularization has become an important tool for solving many ill-posed problems in approximation theory--for example, in computer vision--including surface reconstruction, optical flow, and shape from shading. The authors attempt to determine whether the approach taken in regularization is always the correct one, and to what extent the results of regularization are reliable. For example, the authors consider a case in which regularization has been used to reconstruct a surface from sparse data and attempt to determine how strongly the height of the surface at a certain point can be relied upon. These questions are answered by defining a probability distribution on the class of surfaces considered, and computing its expectation and variance. The variance can be used, for instance, to construct a safety strip around the interpolated surface that should not be entered if collision with the surface is to be avoided.