Gaussian process modeling is a standard tool for building emulators for computer experiments, which are usually used to study deterministic functions, for example, a solution to a given system of partial differential equations. This work investigates applying Gaussian process modeling to a deterministic function from prediction and uncertainty quantification perspectives, where the Gaussian process model is misspecified. Specifically, we consider the case where the underlying function is fixed and from a reproducing kernel Hilbert space generated by some kernel function, and the same kernel function is used in the Gaussian process modeling as the correlation function for prediction and uncertainty quantification. While upper bounds and optimal convergence rate of prediction in the Gaussian process modeling have been extensively studied in the literature, a thorough exploration of convergence rates and theoretical study of uncertainty quantification is lacking. We prove that, if one uses maximum likelihood estimation to estimate the variance in Gaussian process modeling, under different choices of the nugget parameter value, the predictor is not optimal and/or the confidence interval is not reliable. In particular, lower bounds of the prediction error under different choices of the nugget parameter value are obtained. The results indicate that, if one directly applies Gaussian process modeling to a fixed function, the reliability of the confidence interval and the optimality of the predictor cannot be achieved at the same time.
翻译:高斯进程模型是一种标准工具,用于为计算机实验建立模拟器,通常用于研究确定性功能,例如,部分差异方程式的解决方案。这项工作调查从预测和不确定性量化模型的角度,将高斯进程模型应用到从预测和不确定性量化模型的确定性功能,而高斯进程模型的描述错误。具体地说,我们考虑了以下案例:基础功能是固定的,并来自某些内核函数生成的复制核心希尔伯特空间,而同一内核函数在高斯进程模型中作为预测和不确定性量化的关联函数使用。虽然在文献中广泛研究了高斯进程模型预测的上限和最佳趋同率,但彻底探索了趋同率和不确定性量化理论研究缺乏。我们证明,如果在对纳格特参数值的不同选择下,使用最有可能估计高斯进程模型模型模型模型差异的估算,预测器不是最理想的,而且/或信任间隔不可靠。特别是,高斯模型的上限值无法直接定位,那么,如果在模型下,一个最精确的参数的精确度,则表明在不同的精确度模型下,则无法直接选择。