Improper Priors, Spline Smoothing and the Problem of Guarding Against Model Errors in Regression

SUMMARY Spline and generalized spline smoothing is shown to be equivalent to Bayesian estimation with a partially improper prior. This result supports the idea that spline smoothing is a natural solution to the regression problem when one is given a set of regression functions but one also wants to hedge against the possibility that the true model is not exactly in the span of the given regression functions. A natural measure of the deviation of the true model from the span of the regression functions comes out of the spline theory in a natural way. An appropriate value of this measure can be estimated from the data and used to constrain the estimated model to have the estimated deviation. Some convergence results and computational tricks are also discussed.