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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.

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They are completely different things. You could certainly use Gaussian process regression to fit a GP to the same data to which you could also try to fit a GMM. One major difference is that a Gaussian process in non-parametric whereas the GMM is parametric.

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