The Average Posterior Variance of a Smoothing Spline and a Consistent Estimate of the Average Squared Error

Abstract

A smoothing spline estimator can be interpreted in two ways: either as the solution to a variational problem or as the posterior mean when a particular Gaussian prior is placed on the unknown regression function. In order to explain the remarkable performance of her Bayesian "confidence intervals" in a simulation study, Wahba conjectured that the average posterior variance of a spline estimate evaluated at the observation points will be close to the expected average squared error. The estimate of the average posterior variance proposed by Wahba is shown to converge in probability to a quantity proportional to the expected average squared error. This result is established by relating this statistic to a consistent risk estimate based on generalized cross-validation.