Gaussian Process Regression (GPR) is a non-parametric Bayesian regression method that provides both predictions and uncertainty estimates — modeling the process response as a sample from a Gaussian process, with the kernel function encoding assumptions about smoothness and correlation structure.

How GPR Works

• Prior: Define a GP prior with mean function and kernel (e.g., squared exponential, Matérn).
• Conditioning: Given observed data, compute the posterior GP (mean = prediction, variance = uncertainty).
• Prediction: New points predicted with mean and confidence intervals.
• Hyperparameters: Kernel parameters are optimized by maximizing the marginal likelihood.

Why It Matters

• Uncertainty Quantification: Every prediction comes with a confidence interval — critical for risk-aware optimization.
• Bayesian Optimization: GPR is the default surrogate model for Bayesian optimization of expensive processes.
• Small Data: Excellent performance with limited data (10-100 observations) — typical for DOE.

GPR is the probabilistic process model — predicting not just the best estimate but how uncertain that estimate is.