Gaussian process regression (GPR) is a popular nonparametric Bayesian method that provides predictive uncertainty estimates and is widely used in safety-critical applications. While prior research has introduced various uncertainty bounds, most existing approaches require access to specific input features, and rely on posterior mean and variance estimates or the tuning of hyperparameters. These limitations hinder robustness and fail to capture the model's global behavior in expectation. To address these limitations, we propose a chaining-based framework for estimating upper and lower bounds on the expected extreme values over unseen data, without requiring access to specific input features. We provide kernel-specific refinements for commonly used kernels such as RBF and Matérn, in which our bounds are tighter than generic constructions. We further improve numerical tightness by avoiding analytical relaxations. In addition to global estimation, we also develop a novel method for local uncertainty quantification at specified inputs. This approach leverages chaining geometry through partition diameters, adapting to local structures without relying on posterior variance scaling. Our experimental results validate the theoretical findings and demonstrate that our method outperforms existing approaches on both synthetic and real-world datasets.

翻译：高斯过程回归（GPR）是一种流行的非参数贝叶斯方法，能够提供预测不确定性估计，广泛应用于安全关键型任务。尽管先前研究已提出多种不确定性界，但现有方法大多需要依赖特定输入特征，并基于后验均值与方差估计或超参数调优。这些限制影响了方法的鲁棒性，且无法在期望意义上捕捉模型的全局行为。为解决上述问题，我们提出一种基于链式方法的框架，用于估计未见数据上期望极值的上下界，无需访问特定输入特征。针对常用核函数（如RBF核与Matérn核），我们提供了核函数特化的改进方案，所得界比通用构造更紧。通过避免解析松弛，我们进一步提升了数值紧度。除全局估计外，我们还开发了一种在指定输入处进行局部不确定性量化的新方法。该方法通过划分直径利用链式几何结构，能够适应局部特征且不依赖后验方差缩放。实验结果验证了理论结论，并表明我们的方法在合成与真实数据集上均优于现有方法。