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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 $\mathcal{G}_f \cdot \mathcal{G}_g$?

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I think that compositions of Gaussian Processes are called Deep Gaussian Processes.

Typically, I guess the way a composition $f \circ g$ would be defined is, if both functions are defined on $\mathbb R$, then $f \circ g(x) \equiv f(g(x))$.

So, essentially, if both functions were GPs, you'd be calculating:

$$f \sim \mathcal N(0, \Sigma_f(g(x)))$$ $$g \sim \mathcal N(0, \Sigma_g(x))$$

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.

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