Mathematical models implemented on a computer have become the driving force behind the acceleration of the cycle of scientific processes. This is because computer models are typically much faster and economical to run than physical experiments. In this work, we develop an empirical Bayes approach to predictions of physical quantities using a computer model, where we assume that the computer model under consideration needs to be calibrated and is computationally expensive. We propose a Gaussian process emulator and a Gaussian process model for the systematic discrepancy between the computer model and the underlying physical process. This allows for closed-form and easy-to-compute predictions given by a conditional distribution induced by the Gaussian processes. We provide a rigorous theoretical justification of the proposed approach by establishing posterior consistency of the estimated physical process. The computational efficiency of the methods is demonstrated in an extensive simulation study and a real data example. The newly established approach makes enhanced use of computer models both from practical and theoretical standpoints.

在计算机上实现的数学模型已成为推动科学进程循环加速的驱动力。这是因为计算机模型通常比物理实验运行起来更快、更经济。在这项工作中，我们开发了一种使用计算机模型预测物理量的经验贝叶斯方法，我们假设所考虑的计算机模型需要校准并且计算成本很高。针对计算机模型与底层物理过程之间的系统差异，我们提出了高斯过程模拟器和高斯过程模型。这允许由高斯过程引起的条件分布给出的封闭形式和易于计算的预测。我们通过建立估计的物理过程的后验一致性，为所提出的方法提供了严格的理论依据。这些方法的计算效率在广泛的模拟研究和真实数据示例中得到了证明。新建立的方法从实践和理论的角度增强了计算机模型的使用。