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、函数逼近及机器学习中的高斯过程方法。其广受欢迎的原因之一在于平滑度参数的存在——该参数控制着高斯过程回归中的克里金最优误差界和后验收缩率等关键指标。本文证明，在闭Riemannian流形上，当基础数据来自高斯过程的点评估时，甚至出人意料地当数据包含非高斯过程的评估时，该平滑度参数均可通过高斯似然的最大化（或多个极大值点）实现一致估计。除准均匀性条件外，过程观测点无需具备任何特殊空间结构。我们的方法基于Sobolev尺度Hilbert空间的逼近理论结果。此外，通过运用角谷定理，我们将与Matérn核相关的测度等价经典现象推广至非高斯情形。