Bayesian Projected Calibration of Computer Models

We develop a Bayesian approach called the Bayesian projected calibration to address the problem of calibrating an imperfect computer model using observational data from an unknown complex physical system. The calibration parameter and the physical system are parameterized in an identifiable fashion via the L2-projection. The physical system is imposed a Gaussian process prior distribution, which naturally induces a prior distribution on the calibration parameter through the L2-projection constraint. The calibration parameter is estimated through its posterior distribution, serving as a natural and nonasymptotic approach for the uncertainty quantification. We provide rigorous large sample justifications of the proposed approach by establishing the asymptotic normality of the posterior of the calibration parameter with the efficient covariance matrix. In addition to the theoretical analysis, two convenient computational algorithms based on stochastic approximation are designed with strong theoretical support. Through extensive simulation studies and the analyses of two real-world datasets, we show that the proposed Bayesian projected calibration can accurately estimate the calibration parameters, calibrate the computer models well, and compare favorably to alternative approaches. Supplementary materials for this article are available online.

本研究提出一种名为贝叶斯投影校准（Bayesian projected calibration）的贝叶斯方法，用于利用未知复杂物理系统的观测数据，对存在缺陷的计算机模型开展校准工作。校准参数与物理系统通过L2投影（L2-projection）以可识别的方式完成参数化。我们为物理系统赋予高斯过程（Gaussian process）先验分布，该分布可通过L2投影约束自然诱导出校准参数的先验分布。通过校准参数的后验分布对其进行估计，该方法可作为一种自然且非渐近的不确定性量化（uncertainty quantification）手段。本研究通过证明校准参数后验分布的渐近正态性，并推导得到其高效协方差矩阵，为所提方法提供了严谨的大样本理论支撑。除理论分析外，本研究还设计了两种基于随机近似（stochastic approximation）的便捷计算算法，并为其提供了坚实的理论依据。通过大量仿真实验与两个真实世界数据集的分析，本研究证明所提的贝叶斯投影校准方法能够精准估计校准参数，有效完成计算机模型校准，且相较于其他替代方法具有更优的性能。本文的补充材料可在线获取。