# Is there such a thing as composition of two Gaussian Processes?

I have a very basic understanding of Gaussian Processes. From what I understand, a Guassian process for a set $$X$$, is the assignment of a Gaussian distribution to every element of the set. This is meant to expand the idea of a function to the case where we don't have total information about a function.

Two functions, $$f$$ and $$g$$, have a natural notion of composition $$f \cdot g$$, ie just function composition. If Gaussian Processes are approximations to functions, is there a notion of composition of two Gaussian Processes? Given two Gaussian Processes, $$\mathcal{G}_f, \mathcal{G}_g$$, how is the composition defined $$\mathcal{G}_f \cdot \mathcal{G}_g$$?

Typically, I guess the way a composition $$f \circ g$$ would be defined is, if both functions are defined on $$\mathbb R$$, then $$f \circ g(x) \equiv f(g(x))$$.
$$f \sim \mathcal N(0, \Sigma_f(g(x)))$$ $$g \sim \mathcal N(0, \Sigma_g(x))$$
Given that the covariance matrix of $$f$$ depends on the values $$g$$, I do not think that there exists any nice closed form descriptions of this process, although a literature search for deep GPs might prove me wrong.