高斯过程插值中的参数选择：选择准则的实证研究
Parameter selection in Gaussian process interpolation: an empirical study of selection criteria
AI总结：
本文基于评分规则和扩展似然准则，实证研究高斯过程插值中的参数选择问题，发现模型族的选择通常比具体选择准则更重要，且大多数准则都能有效选取Matérn协方差的正则性参数。
摘要：
本文重新审视高斯过程插值中参数选择这一基本问题。通过在参数族中选择高斯过程的均值函数和协方差函数，用户可以获得一族用于对未知函数进行预测的贝叶斯方法，并且必须从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（scoring rules）这一一般概念，它为构建留一法（leave-one-out）选择准则和验证准则提供了有效框架；同时基于Fasshauer及其合作者在2009年提出的思想，引入一种扩展似然准则（extended likelihood criteria）的概念，使得可以恢复诸如广义交叉验证（generalized cross-validation）准则等标准选择准则。在此设定下，我们在文献中的多个测试问题上通过实证表明，选择合适的模型族往往比选择某个特定的选择准则（例如，似然准则与留一法选择准则之间的比较）更为重要。此外，数值结果表明，Matérn协方差的正则性参数可以通过大多数选择准则被有效选取。
英文摘要：
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{é}rn covariance can be selected effectively by most selection criteria.
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