Asymptotic properties of Vecchia approximation for Gaussian processes

TL;DR

This paper analyzes the asymptotic properties of Vecchia approximation for Gaussian processes, demonstrating its theoretical equivalence to exact inference under certain conditions and its practical advantages over other methods.

Contribution

It establishes the asymptotic normality and consistency of Vecchia-based inference for a broad class of covariance functions, including boundary conditioning and large data sets.

Findings

Vecchia approximation achieves asymptotic equivalence to exact GP inference.

It can be more accurate than covariance tapering and reduced-rank methods.

The method has quasilinear complexity and reliable prediction accuracy.

Abstract

Vecchia approximation has been widely used to accurately scale Gaussian-process (GP) inference to large datasets, by expressing the joint density as a product of conditional densities with small conditioning sets. We study fixed-domain asymptotic properties of Vecchia-based GP inference for a large class of covariance functions (including Mat\'ern covariances) with boundary conditioning. In this setting, we establish that consistency and asymptotic normality of maximum exact-likelihood estimators imply those of maximum Vecchia-likelihood estimators, and that exact GP prediction can be approximated accurately by Vecchia GP prediction, given that the size of conditioning sets grows polylogarithmically with the data size. Hence, Vecchia-based inference with quasilinear complexity is asymptotically equivalent to exact GP inference with cubic complexity. This also provides a general new result on the screening effect. Our findings are illustrated by numerical experiments, which also show that Vecchia approximation can be more accurate than alternative approaches such as covariance tapering and reduced-rank approximations.