GaussianProzesse

GaussianProzesse, often described in English as Gaussian processes, refer to a class of stochastic processes used in probability theory and statistics. A GaussianProzesse is a collection {X(t): t in T} of random variables such that for every finite subset t1, ..., tn in T, the random vector (X(t1), ..., X(tn)) follows a multivariate normal distribution. The distribution is completely specified by its mean function m(t) = E[X(t)] and its covariance function k(s, t) = Cov(X(s), X(t)).

Because all finite-dimensional distributions are Gaussian, the entire process is determined by m and k. In particular, for any finite n, the joint distribution is Gaussian with mean vector [m(t1), ..., m(tn)] and covariance matrix [k(ti, tj)]. GaussianProzesse are not constrained to be stationary, but stationary Gaussian processes have a constant mean and a covariance that depends only on the time difference. The smoothness of sample paths is driven by the covariance kernel; some kernels yield differentiable paths, others yield rough paths.

Common examples include the standard Brownian motion (Wiener process), with m(t) = 0 and k(s, t) = min(s, t), and the Ornstein–Uhlenbeck process, which is Gaussian with an exponential-type covariance. In machine learning, Gaussian Prozesses provide a framework for Gaussian process regression, where a prior over functions is specified by a mean function and a covariance kernel and the posterior given Gaussian noise remains Gaussian. In spatial statistics, they underpin kriging and related interpolation methods. Existence of a GaussianProzesse with given finite-dimensional distributions is guaranteed by general existence results such as the Kolmogorov extension theorem.

Key notation: X(t) for t in T, m(t) for the mean, k(s, t) for the covariance kernel. GaussianProzesse are valued for analytical tractability and flexible covariance modeling.