  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.
