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

翻译：本文重新审视了高斯过程插值中的参数选择这一基本问题。通过从参数族中选择高斯过程的均值函数和协方差函数，用户可以获得一系列贝叶斯程序来对未知函数进行预测，并且必须从该族中选择一个有望提供良好预测性能的成员。我们的研究基于评分规则的一般概念，该概念为构建留一法选择和验证标准提供了一个有效框架，并基于Fasshauer及其合作者于2009年提出的一种思想，提出了扩展似然标准的概念，从而能够恢复标准选择标准（例如广义交叉验证标准）。在这一框架下，我们针对文献中的多个测试问题通过实证表明，选择适当的模型族通常比选择特定的选择标准（例如，似然法与留一法选择标准）更为重要。此外，我们的数值结果表明，大多数选择标准能够有效地选择Matérn协方差的正则性参数。