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This article revisits the fundamental problem of parameter selection forGaussian process interpolation. By choosing the mean and the covariancefunctions of a Gaussian process within parametric families, the user obtains afamily of Bayesian procedures to perform predictions about the unknownfunction, and must choose a member of the family that will hopefully providegood predictive performances. We base our study on the general concept ofscoring rules, which provides an effective framework for building leave-one-outselection and validation criteria, and a notion of extended likelihood criteriabased on an idea proposed by Fasshauer and co-authors in 2009, which makes itpossible to recover standard selection criteria such as, for instance, thegeneralized cross-validation criterion. Under this setting, we empirically showon several test problems of the literature that the choice of an appropriatefamily of models is often more important than the choice of a particularselection criterion (e.g., the likelihood versus a leave-one-out selectioncriterion). Moreover, our numerical results show that the regularity parameterof a Mat{\'e}rn covariance can be selected effectively by most selectioncriteria.

Parameter selection in Gaussian process interpolation: an empirical study of selection criteria