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.

翻译：维基亚近似通过将联合密度表示为具有小条件集的条件密度乘积，已广泛应用于将高斯过程（GP）推断准确扩展到大规模数据集。我们研究了一类具有边界条件协方差函数（包括Matérn协方差）的维基亚GP推断的固定域渐近性质。在此设定下，我们证明了当条件集大小随数据量呈多对数增长时，最大精确似然估计量的一致性及渐近正态性可推导出最大维基亚似然估计量的相应性质，且精确GP预测可被维基亚GP预测准确逼近。因此，具有拟线性复杂度的维基亚推断与三次复杂度的精确GP推断在渐近意义下等价。这同时为屏蔽效应提供了新的普适性结论。数值实验验证了我们的发现，并表明维基亚近似比协方差削尖和降秩近似等替代方法具有更高精度。