# Gaussian process

A Gaussian process is a stochastic process {Xt}tT such that every finite linear combination of the Xt (or, more generally, any linear functional of the sample function Xt) is normally distributed. The concept is named after Carl Friedrich Gauss simply because the normal distribution is sometimes called the Gaussian distribution, although Gauss was not the first to study that distribution. Note that some authors (for example B. Simon in the reference cited below) also assume the variables Xt have mean zero. Alternatively, a process is Gaussian iff for every finite set of indices t1, ..., tk in the index set T

[itex] \vec{\mathbf{X}}_{t_1, \ldots, t_k} = (\mathbf{X}_{t_1}, \ldots, \mathbf{X}_{t_k}) [itex]

is a vector-valued Gaussian random variable. Using characteristic functions of random variables, we can formulate the Gaussian property as follows:{Xt}tT is Gaussian iff for every finite set of indices t1, ..., tk there are positive reals σl j and reals μj such that

[itex] \operatorname{E}\left(\exp\left(i \ \sum_{\ell=1}^k t_\ell \ \mathbf{X}_{t_\ell}\right)\right) = \exp \left(-\frac{1}{2} \, \sum_{\ell, j} \sigma_{\ell j} t_\ell t_j + i \sum_\ell \mu_\ell t_\ell\right). [itex]

The numbers σl j and μj can be shown to be the covariances and means of the variables in the process.

The Wiener process is perhaps the most widely studied Gaussian process.

A Gaussian process can be used as a prior probability distribution over functions in Bayesian inference. Inference of continuous values with a Gaussian process prior is known as Gaussian process regression.

## References

• R. M. Dudley, Real Analysis and Probability, Wadsworth and Brooks/Cole, 1989.
• B. Simon, Functional Integration and Quantum Physics, Academic Press, 1979.

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