The Gaussian process (GP) regression model is a widely employed surrogate modeling technique for computer experiments, offering precise predictions and statistical inference for the computer simulators that generate experimental data. Estimation and inference for GP can be performed in both frequentist and Bayesian frameworks. In this chapter, we construct the GP model through variational inference, particularly employing the recently introduced energetic variational inference method by Wang et al. (2021). Adhering to the GP model assumptions, we derive posterior distributions for its parameters. The energetic variational inference approach bridges the Bayesian sampling and optimization and enables approximation of the posterior distributions and identification of the posterior mode. By incorporating a normal prior on the mean component of the GP model, we also apply shrinkage estimation to the parameters, facilitating mean function variable selection. To showcase the effectiveness of our proposed GP model, we present results from three benchmark examples.

翻译：高斯过程（GP）回归模型是一种广泛应用于计算机实验的代理建模技术，能够为生成实验数据的计算机模拟器提供精确预测与统计推断。GP的估计与推断可在频率学派和贝叶斯学派两种框架下进行。本章通过变分推断构建GP模型，特别采用Wang等（2021）新近提出的能量变分推断方法。在遵循GP模型假设的前提下，我们推导了其参数的后验分布。该能量变分推断方法架起了贝叶斯采样与优化之间的桥梁，能够近似后验分布并识别后验众数。通过将正态先验引入GP模型的均值分量，我们还对参数实施了收缩估计，从而实现了均值函数变量选择。为验证所提GP模型的有效性，我们展示了三个基准案例的实验结果。