Gaussianprocess

Gaussianprocess is a probabilistic model that defines a distribution over functions. In this framework, a function f mapping an input x to a real value is assumed to be drawn from a Gaussian process, which is specified by a mean function m(x) and a covariance function k(x, x'). For any finite set of inputs x1, ..., xn, the function values f(x1), ..., f(xn) follow a multivariate normal distribution with mean vector [m(x1), ..., m(xn)] and covariance matrix [k(xi, xj)].

A common setting introduces observed data y with noise: y_i = f(x_i) + ε_i, where ε_i are independent Gaussian errors with variance σ^2. The prior over function values at the inputs X = (x1, ..., xn) is f(X) ~ N(m(X), K(X,X)) with K(X,X)_{ij} = k(x_i, x_j). The marginal likelihood of the observations is p(y|X) = N(y; m(X), K(X,X) + σ^2 I).

Inference in Gaussian processes yields a Gaussian posterior over function values at any test inputs X* given data X and y. The posterior mean and covariance are:

m*(X*) = m(X*) + K(X*,X) [K(X,X) + σ^2 I]^{-1} (y - m(X))

cov*(X*) = K(X*,X*) - K(X*,X) [K(X,X) + σ^2 I]^{-1} K(X, X*)

These formulas provide predictive distributions for new points, including uncertainty. Kernels encode assumptions about function properties; common choices include the squared exponential (RBF), Matérn, and periodic kernels. Hyperparameters in m and k (and σ) are often learned by maximizing the marginal likelihood or treated in a fully Bayesian way.

Gaussian processes are used in regression, time-series analysis, spatial statistics, Bayesian optimization, and surrogate modeling. They offer flexible, nonparametric modeling with principled uncertainty quantification, while computational challenges arise for large datasets, motivating scalable approaches such as sparse or inducing-point methods.