I am familiarizing myself with multivariate Gaussian distributions and Gaussian processes using
george, and I am trying to understand how sampling from a multivariate Gaussian distribution works.
The following is a sample from the prior of a multivariate Gaussian distribution, using
numpy.random.multivariate_normal. Each value of x on the graph corresponds to one dimension in the distribution - the graph was generated with 500 values of x equally spaced between 0 and 10. The multivariate Gaussian distribution was generated with mean at 0.0 across all 500 dimensions, and a covariance matrix generated using a squared exponential kernel.
As can be seen above, the sample appears "continuous".
My understanding is that adjacent points on the graph (corresponding to the values sampled from each dimension) are related to each other purely by the covariance matrix - so the fact that the sample looks like a continuous function stems from the covariance causing values in "adjacent" dimensions (?) to be closer to each other. Is my understanding correct, or is this because of the particular way that
numpy samples from a
multivariate_normal, or maybe because of some other reasons?