The family of Mat\'ern kernels are often used in spatial statistics, function approximation and Gaussian process methods in machine learning. One reason for their popularity is the presence of a smoothness parameter that controls, for example, optimal error bounds for kriging and posterior contraction rates in Gaussian process regression. On closed Riemannian manifolds, we show that the smoothness parameter can be consistently estimated from the maximizer(s) of the Gaussian likelihood when the underlying data are from point evaluations of a Gaussian process and, perhaps surprisingly, even when the data comprise evaluations of a non-Gaussian process. The points at which the process is observed need not have any particular spatial structure beyond quasi-uniformity. Our methods are based on results from approximation theory for the Sobolev scale of Hilbert spaces. Moreover, we generalize a well-known equivalence of measures phenomenon related to Mat\'ern kernels to the non-Gaussian case by using Kakutani's theorem.

翻译：Matérn核函数族常用于空间统计学、函数逼近以及机器学习中的高斯过程方法。其广泛应用的缘由之一在于其包含一个平滑度参数，该参数可控制如克里金法的最优误差界和高斯过程回归中的后验收缩速率。在闭黎曼流形上，我们证明：当底层数据来自高斯过程的点观测值时，该平滑度参数可通过高斯似然函数的极大值点得到一致估计；令人意外的是，即使数据包含非高斯过程的评估值，该结论依然成立。观测过程的采样点除需拟均匀性外，无需具备特定空间结构。我们的方法基于索伯列夫空间尺度下的逼近理论成果。此外，我们通过运用角谷定理，将Matérn核相关的著名测度等价现象推广至非高斯情形。