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. \citet{ma2022beyond} 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 for nonparametric regression model with fixed design. 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. We also consider a fully Bayesian treatment of the rescaling parameters and show the resulting posterior distributions still contract at the minimax-optimal rate. The resultant hierarchical Bayesian procedure is fully adaptive to the unknown true smoothness.

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率的文献表明，若高斯过程的平滑度与目标函数相匹配，则可达到最优或接近最优的后验收缩速率。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，因其平滑度可通过专用平滑度参数确定。\citet{ma2022beyond}近期引入了一类称为合流超几何（CH）族的新协方差函数，其同时具备两个参数：一个控制多项式衰减协方差函数的尾部指数，另一个参数控制与Matérn类类似的均方平滑度。本文证明，通过适当选择Matérn和CH协方差函数中的重标度参数，对于固定设计非参数回归模型中的$\eta$正则函数，可获得极小极大最优的后验收缩速率。与未重标度情形的先前结果不同，无论对于重标度Matérn还是重标度CH协方差，为达到最优极小极大速率，协方差函数的平滑度参数均无需等于$\eta$，这揭示了重标度的关键优势。我们还对重标度参数进行了完全贝叶斯处理，证明所得后验分布仍以极小极大最优速率收缩。由此产生的分层贝叶斯程序能完全自适应于未知的真实平滑度。