gaussian process regression

**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.