The surrogate places a Gaussian process prior over f with an RBF (squared-exponential) covariance k(x,x') = σf² exp(-(x-x')²/2ℓ²). Given n noisy observations y = f(X) + ε with ε ~ N(0, σε²), the posterior mean and variance at a new point x* follow from the standard GP regression equations, solved here via a Cholesky factorization of K + σε²I rather than an explicit matrix inverse — the numerically stable route used in every production GP library.
Expected improvement (EI) integrates the improvement over the current best f_best against the posterior: with Z = (μ(x) − f_best − ξ)/σ(x), EI(x) = (μ(x) − f_best − ξ)Φ(Z) + σ(x)φ(Z). Probability of improvement (PI) keeps only the Φ(Z) term — it rewards any improvement, however small, and is consequently more exploitative. Upper confidence bound (UCB), μ(x) + κσ(x), trades off explicitly via κ with no reference to f_best at all.
k(x,x') = \sigma_f^2 \exp\!\left(-\frac{(x-x')^2}{2\ell^2}\right)\mu(x_*) = k_*^\top (K+\sigma_\varepsilon^2 I)^{-1} y, \quad \sigma^2(x_*) = k(x_*,x_*) - k_*^\top (K+\sigma_\varepsilon^2 I)^{-1} k_*\mathrm{EI}(x) = (\mu(x)-f_{best}-\xi)\Phi(Z) + \sigma(x)\phi(Z), \quad Z=\frac{\mu(x)-f_{best}-\xi}{\sigma(x)}