Editing Gaussian process (section)

==Linearly constrained Gaussian processes==
For many applications of interest some pre-existing knowledge about the system at hand is already given. Consider e.g. the case where the output of the Gaussian process corresponds to a magnetic field; here, the real magnetic field is bound by Maxwell's equations and a way to incorporate this constraint into the Gaussian process formalism would be desirable as this would likely improve the accuracy of the algorithm.

A method on how to incorporate linear constraints into Gaussian processes already exists:<ref>{{cite arXiv| last1=Jidling|first1=Carl| last2=Wahlström|first2=Niklas|last3=Wills|first3=Adrian|last4=Schön|first4=Thomas B.| date=2017-09-19| title=Linearly constrained Gaussian processes|class=stat.ML|eprint=1703.00787}}</ref>

Consider the (vector valued) output function <math>f(x)</math> which is known to obey the linear constraint (i.e. <math>\mathcal{F}_X</math> is a linear operator)
<math display="block">\mathcal{F}_X(f(x)) = 0.</math>
Then the constraint <math>\mathcal{F}_X</math> can be fulfilled by choosing <math>f(x) = \mathcal{G}_X(g(x))</math>, where <math>g(x) \sim \mathcal{GP}(\mu_g, K_g)</math> is modelled as a Gaussian process, and finding <math>\mathcal{G}_X</math> such that 
<math display="block">\mathcal{F}_X(\mathcal{G}_X(g)) = 0 \qquad \forall g.</math>
Given <math>\mathcal{G}_X</math> and using the fact that Gaussian processes are closed under linear transformations, the Gaussian process for <math>f</math> obeying constraint <math>\mathcal{F}_X</math> becomes
<math display="block">f(x) = \mathcal{G}_X g \sim \mathcal{GP} ( \mathcal{G}_X \mu_g, \mathcal{G}_X K_g \mathcal{G}_{X'}^\mathsf{T} ).</math>
Hence, linear constraints can be encoded into the mean and covariance function of a Gaussian process.