Gaussian Process Learning via Fisher Scoring of Vecchia's Approximation

The Gaussian process model is an indispensible tool for the analysis of spatial and spatial-temporal datasets and has become increasingly popular as a general-purpose model for functions. Because of its high computational burden, researchers have devoted substantial effort to developing numerical approximations for Gaussian process computations. Much of the work focuses on efficient approximation of the likelihood function. Fast likelihood evaluations are crucial for optimization procedures that require many evaluations of the likelihood, such as the default Nelder-Mead algorithm (Nelder and Mead, 1965) in the R optim function. The likelihood must be repeatedly evaluated in MCMC algorithms as well. Compared to the amount of literature on efficient likelihood approximations, there has been considerably less development of techniques for numerically maximizing the likelihood (see Geoga et al. (2018) for one recent example). This article aims to address the disparity by providing: 1. Formulas for evaluating the gradient and Fisher information for Vecchia's likelihood approximation in a single pass through the data, so that the Fisher scoring algorithm can be applied. Fisher scoring is a modification of the Newton-Raphson optimization method, replacing the Hessian matrix with the Fisher information matrix.