Gaussian Process Regression (GPR) is widely used in statistics and machine learning for prediction tasks requiring uncertainty measures. Its efficacy depends on the appropriate specification of the mean function, covariance kernel function, and associated hyperparameters. Severe misspecifications can lead to inaccurate results and problematic consequences, especially in safety-critical applications. However, a systematic approach to handle these misspecifications is lacking in the literature. In this work, we propose a general framework to address these issues. Firstly, we introduce a flexible two-stage GPR framework that separates mean prediction and uncertainty quantification (UQ) to prevent mean misspecification, which can introduce bias into the model. Secondly, kernel function misspecification is addressed through a novel automatic kernel search algorithm, supported by theoretical analysis, that selects the optimal kernel from a candidate set. Additionally, we propose a subsampling-based warm-start strategy for hyperparameter initialization to improve efficiency and avoid hyperparameter misspecification. With much lower computational cost, our subsampling-based strategy can yield competitive or better performance than training exclusively on the full dataset. Combining all these components, we recommend two GPR methods-exact and scalable-designed to match available computational resources and specific UQ requirements. Extensive evaluation on real-world datasets, including UCI benchmarks and a safety-critical medical case study, demonstrates the robustness and precision of our methods.

翻译：高斯过程回归（GPR）在统计学和机器学习中被广泛用于需要不确定性度量的预测任务。其有效性取决于均值函数、协方差核函数及相关超参数的恰当设定。严重的设定错误可能导致不准确的结果及严重后果，尤其在安全关键型应用中。然而，现有文献缺乏处理此类设定错误的系统性方法。本文提出一个通用框架以解决这些问题。首先，我们引入一种灵活的两阶段GPR框架，将均值预测与不确定性量化（UQ）分离，以避免均值设定错误可能引入的模型偏差。其次，通过一种基于理论分析的新型自动核搜索算法，从候选集中选择最优核函数，以解决核函数设定错误问题。此外，我们提出一种基于子采样的热启动策略用于超参数初始化，以提高效率并避免超参数设定错误。该子采样策略能以远低于全数据集训练的计算成本，获得与之相当或更优的性能。综合以上组件，我们推荐两种GPR方法——精确型与可扩展型，以适应不同的计算资源与特定UQ需求。在包括UCI基准数据集和安全关键型医学案例研究在内的真实数据集上的广泛评估，证明了我们方法的鲁棒性与精确性。