In nonparameteric Bayesian approaches, Gaussian stochastic processes can serve as priors on real-valued function spaces. Existing literature on the posterior convergence rates under Gaussian process priors shows that it is possible to achieve optimal or near-optimal posterior contraction rates if the smoothness of the Gaussian process matches that of the target function. Among those priors, Gaussian processes with a parametric Mat\'ern covariance function is particularly notable in that its degree of smoothness can be determined by a dedicated smoothness parameter. Ma and Bhadra(2023) recently introduced a new family of covariance functions called the Confluent Hypergeometric (CH) class that simultaneously possess two parameters: one controls the tail index of the polynomially decaying covariance function, and the other parameter controls the degree of mean-squared smoothness analogous to the Mat\'ern class. In this paper, we show that with proper choice of rescaling parameters in the Mat\'ern and CH covariance functions, it is possible to obtain the minimax optimal posterior contraction rate for $\eta$-regular functions. Unlike the previous results for unrescaled cases, the smoothness parameter of the covariance function need not equal $\eta$ for achieving the optimal minimax rate, for either rescaled Mat\'ern or rescaled CH covariances, illustrating a key benefit for rescaling. The theoretical properties of the rescaled Mat\'ern and CH classes are further verified via extensive simulations and an illustration on a geospatial data set is presented.

翻译：在非参数贝叶斯方法中，高斯随机过程可作为实值函数空间上的先验分布。现有关于高斯过程先验下后验收敛速度的研究表明，当高斯过程的平滑性与目标函数相匹配时，可实现最优或接近最优的后验收缩率。在这些先验中，具有参数化 Matérn 协方差函数的高斯过程尤为突出，因其平滑程度可由专用平滑参数确定。Ma 和 Bhadra（2023）近期提出了一类新型协方差函数——合流超几何（CH）类，该类函数同时具备两个参数：一个控制多项式衰减协方差函数的尾指数，另一个控制类似于 Matérn 类的均方平滑程度。本文证明：通过对 Matérn 和 CH 协方差函数适当选择重缩放参数，可获得 $\eta$-正则函数的极小极大最优后验收缩率。与未重缩放情形下的先前结果不同，对于重缩放 Matérn 或重缩放 CH 协方差，实现最优极小极大率时协方差函数的平滑参数无需等于 $\eta$，这凸显了重缩放的关键优势。通过大量仿真实验进一步验证了重缩放 Matérn 和 CH 类的理论性质，并展示了在地理空间数据集上的应用实例。