# Reducibility between Gaussian Mixture Models and Gaussian Processes

I am studying gaussian processes and I have already discrete amount of knowledge in gaussian mixture models. I am here to undersrtand if with a gaussian process you can fit a gaussian mixture model.

Formally, a GMM is a linear combination of gaussians such that $$\phi(x) = \sum_{i=0}^k \alpha_i \phi_i(x | \mu_i, \Sigma_i)$$ where each $$\phi_i$$ is a gaussian centered in $$\mu_i$$ with variance $$\Sigma_i$$. Computationally this is solved using EM.

A GP is (roughly) a set of functions distributed with a multivariate gaussian probability distribution that models your data. Computationally this is solved by Cholesky decomposition and linear systems.

So I am wondering if with GP you can hope to solve GMM models, or if there is a link whatsoever. To me, they are two completely different things. Am I right? Thanks.