Derivatives are a key nonparametric functional in wide-ranging applications where the rate of change of an unknown function is of interest. In the Bayesian paradigm, Gaussian processes (GPs) are routinely used as a flexible prior for unknown functions, and are arguably one of the most popular tools in many areas. However, little is known about the optimal modelling strategy and theoretical properties when using GPs for derivatives. In this article, we study a plug-in strategy by differentiating the posterior distribution with GP priors for derivatives of any order. This practically appealing plug-in GP method has been previously perceived as suboptimal and degraded, but this is not necessarily the case. We provide posterior contraction rates for plug-in GPs and establish that they remarkably adapt to derivative orders. We show that the posterior measure of the regression function and its derivatives, with the same choice of hyperparameter that does not depend on the order of derivatives, converges at the minimax optimal rate up to a logarithmic factor for functions in certain classes. We analyze a data-driven hyperparameter tuning method based on empirical Bayes, and show that it satisfies the optimal rate condition while maintaining computational efficiency. This article to the best of our knowledge provides the first positive result for plug-in GPs in the context of inferring derivative functionals, and leads to a practically simple nonparametric Bayesian method with optimal and adaptive hyperparameter tuning for simultaneously estimating the regression function and its derivatives. Simulations show competitive finite sample performance of the plug-in GP method. A climate change application for analyzing the global sea-level rise is discussed.

翻译：导数是广泛关注未知函数变化率的应用场景中的关键非参数泛函。在贝叶斯框架下，高斯过程（Gaussian processes, GPs）常被用作未知函数的灵活先验，且在许多领域中堪称最常用的工具之一。然而，当使用高斯过程进行导数建模时，关于最优建模策略及理论特性的研究尚不充分。本文研究了一种插入式策略：对任意阶导数，通过对高斯过程先验的后验分布进行微分来建模。这种具有实际吸引力的插入式高斯过程方法此前被认为并非最优且性能退化，但事实未必如此。我们给出了插入式高斯过程的后验收缩率，并揭示其能显著适应导数阶数。研究表明，当超参数的选择与导数阶数无关时，回归函数及其导数的后验测度对于特定函数类而言，能以接近极大极小最优速率（仅相差对数因子）收敛。我们分析了一种基于经验贝叶斯的数据驱动超参数调优方法，证明其在保持计算效率的同时满足最优速率条件。据我们所知，本文首次为插入式高斯过程在导数泛函推断中提供了正面结果，并由此发展出一种实践简便的非参数贝叶斯方法——通过最优自适应超参数调优同时估计回归函数及其导数。仿真实验展示了插入式高斯过程方法在有限样本下的竞争性能。我们还讨论了分析全球海平面上升的气候变化应用案例。