8.3 Multivariate Gaussians and functions

In Figure 8.1, we represented a function as a collection of samples from 1-dimensional Gaussian distributions. One alternative is to use an n-dimensional multivariate Gaussian distribution to get a sample vector of length n. Actually, you may want to try to reproduce Figure 8.1 but replacing np.random.normal(0, 1, len(x)) with np.random.multivariate_normal, with a mean of np.zeros_like(x) and a standard deviation of np.eye(len(x). The advantage of working with a Multivariate Normal is that we can use the covariance matrix to encode information about the function. For instance, by setting the covariance matrix to np.eye(len(x)), we are saying that each of the 10 points, where we are evaluating the function, has a variance of 1. We are also saying that the variance between them, that is, their covariances, is 0. In other words, they are independent. If we replace those zeros with other numbers, we could get covariances telling a different story...