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）回归模型是计算机实验中广泛采用的代理建模技术，能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模型的估计与推断可在频率学派和贝叶斯学派框架下实施。本章通过变分推断构建GP模型，特别采用Wang等人（2021）近期提出的能量变分推断方法。依据GP模型假设，我们推导了其参数的后验分布。能量变分推断方法桥接了贝叶斯采样与优化，能够近似后验分布并识别后验众数。通过在GP模型的均值分量上引入正态先验，我们对参数施加收缩估计，从而实现均值函数的变量选择。为验证所提GP模型的有效性，我们展示了三个基准测试案例的结果。