Is Cp an empirical Bayes method for smoothing parameter choice?

The Cp selection criterion is a popular method to choose the smoothing parameter in spline regression. Another widely used method is the generalized maximum likelihood (GML) derived from a normal-theory empirical Bayes framework. These two seemingly unrelated methods, have been shown in Efron (Ann. Statist. 29 (2001) 470) and Kou and Efron (J. Amer. Statist. Assoc. 97 (2002) 766) to be actually closely connected. Because of this close relationship, the current paper studies whether Cp could also have an empirical Bayes interpretation for smoothing splines as GML does. It is shown that this is not possible. In addition, necessary conditions for a selection criterion to have an empirical Bayes interpretation are given, using which it is shown that a large class of selection criteria, including Akaike information criterion, Bayesian information criterion and Stein's unbiased risk estimate, does not possess an empirical Bayes explanation.