[Q] understanding Gaussian Processes: how can we assume a jointly distributed multivariate normal distribution?

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  A Visual Exploration of Gaussian Processes

  

Gaussian processes are distributions over functions. Each function value y=f(x) is a dimension, so GPs are infinite dimensional. In practice, we have some finite training and test sets. As the latter quote says, this is just one incomplete sample from the distribution. A single complete sample would be an entire function, ie an infinite dimensional vector. We only observe the function in a finite number of points, but this enough to learn something about the distribution over functions, if you make some prior assumptions about what this distribution is like. Specifically, you would assume that the N+M dimensions are not independent. They are correlated according to the covariance function, which can take many forms. Then conditioning on the N observed points, which is one incomplete sample, gives you a predictive distribution over the test points. This is different from inference in most common models where we have N complete iid samples and a likelihood that is a product of N distributions, and we compute the posterior with some unknown distribution. Instead, we exploit that the multivariate Gaussian has Gaussian conditional and marginal distributions. So we already know the posterior is a multivariate Gaussian, and compute its parameters directly. See https://distill.pub/2019/visual-exploration-gaussian-processes/ for more.

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