Kalman Filtering for Spline Smoothing

In this chapter, we give an account of some Bayesian aspects of spline smoothing for nonparametric regression. The Bayesian view leads to concepts that do not arise in the distinctly non–Bayesian presentation of the previous chapters. Indeed, novel solutions appear to which non–Bayesians cannot object; e.g., the various developments culminating in the Kalman filter for computing spline estimators of arbitrary order m. The efficient computation of the GML and GCV functionals by way of the Kalman filter should not be forgotten. Things already get dicey when justifying the GML method for choosing the smoothing parameter, but it must be admitted that this seems to work very well (see Chapter 23). However, the authors draw the line at Bayesian confidence bands for the unknown regression function.