Gaussian Process Regression

Prior over functions · posterior conditioning · RBF/Matérn kernels · uncertainty bands

Length scale ℓ: 0.50

Signal variance σ²: 1.00

Noise σ²ₙ: 0.10

A GP defines a distribution over functions: f ~ GP(μ, k). The kernel k(x,x') encodes similarity. After observations D = {(xᵢ,yᵢ)}, the posterior is: f*|D ~ GP(μ*, Σ*) where μ* = K*K⁻¹y and Σ* = K** - K*K⁻¹K*ᵀ. The uncertainty band widens where data is sparse. RBF: infinitely differentiable; Matérn 3/2: once-differentiable, rougher.