# Gaussian process estimation

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?

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.