Conventional Gaussian process regression exclusively assumes the existence of noise in the output data of model observations. In many scientific and engineering applications, however, the input locations of observational data may also be compromised with uncertainties owing to modeling assumptions, measurement errors, etc. In this work, we propose a Bayesian method that integrates the variability of input data into Gaussian process regression. Considering two types of observables -- noise-corrupted outputs with fixed inputs and those with prior-distribution-defined uncertain inputs, a posterior distribution is estimated via a Bayesian framework to infer the uncertain data locations. Thereafter, such quantified uncertainties of inputs are incorporated into Gaussian process predictions by means of marginalization. The effectiveness of this new regression technique is demonstrated through several numerical examples, in which a consistently good performance of generalization is observed, while a substantial reduction in the predictive uncertainties is achieved by the Bayesian inference of uncertain inputs.

翻译：传统高斯过程回归仅假设模型观测的输出数据存在噪声。然而在许多科学与工程应用中，由于建模假设、测量误差等原因，观测数据的输入位置也可能存在不确定性。本研究提出一种贝叶斯方法，将输入数据的变异性整合到高斯过程回归中。针对两类可观测量——固定输入下的噪声污染输出与先验分布定义的不确定输入下的输出——通过贝叶斯框架估计后验分布以推断不确定数据位置。随后，通过边缘化方法将这些量化后的输入不确定性纳入高斯过程预测。多个数值算例验证了该新回归技术的有效性：在保持稳定优异泛化性能的同时，通过贝叶斯推断不确定输入显著降低了预测不确定性。