where f = (f (x 1 )), ..., f (x N ) with N the number of samples, µ = (m(x 1 )), ..., m(x N ) and K = k(x, x ′ ). k is a positive definite and symmetric matrix, called kernel function or covariance function (see below for more details) and controls the shape of the distribution over functions. Given a training dataset with values f with inputs X, a GPR prior can be converted into a GPR posterior p(f * , X * , f, X) which can be used to make new predictions about f * with inputs X * . The joint distribution of the observed values f and predicted ones f * can be expressed as a Gaussian: