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The stochastic process $(X_t)_{t\in T}$ is called Gaussian if for all $t_1,\dots,t_k\in T$, for all $k$, the joint distribution of $X_{t_1},\dots,X_{t_k}$ is multivariate normal. The process is completely characterized by its mean $$\mu(t) = E[X_t]$$ and covariance functions $$\sigma(s,t) = Cov[X_s,X_t]$$.

Given a centered (0-mean) Gaussian process, is it possible to estimate its covariance function?

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If the form of the kernel is known (many real applications use an RBF kernel for example), it is possible, given a set of observations $(x_t, y_t)$ to estimate its hyperparameters (the length-scale for RBF) via maximisation of the marginal likelihood.

You should take a look on Chapter 5 of Gaussian Process for Machine Learning. You will find an example of MATLAB code in the gpml documentation, in the paragraph of section "Regression" starting by "Typically, we would not a priori know the values of the hyperparameters..."

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