We study a class of Gaussian processes for which the posterior mean, for a particular choice of data, replicates a truncated Taylor expansion of any order. The data consist of derivative evaluations at the expansion point and the prior covariance kernel belongs to the class of Taylor kernels, which can be written in a certain power series form. We discuss and prove some results on maximum likelihood estimation of parameters of Taylor kernels. The proposed framework is a special case of Gaussian process regression based on data that is orthogonal in the reproducing kernel Hilbert space of the covariance kernel.

翻译：我们研究了一类高斯过程，对于特定数据选择，其后验均值可复现任意截断阶数的泰勒展开。该数据包含展开点处的导数评估，且先验协方差核属于泰勒核类，可表示为特定的幂级数形式。我们讨论并证明了关于泰勒核参数极大似然估计的一些结论。所提出的框架是基于再生核希尔伯特空间中正交数据的高斯过程回归的特例。