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：Vecchia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，已被广泛应用于将高斯过程推断精确扩展到大规模数据集。针对一类含边界条件的协方差函数（包括Matérn协方差），我们研究了基于Vecchia的GP推断在固定域上的渐近性质。在此设定下，我们证明：若最大精确似然估计量具有相合性与渐近正态性，则最大Vecchia似然估计量也具备相同性质；且当条件集大小随数据量呈多对数增长时，精确GP预测可由Vecchia GP预测精确逼近。因此，具有拟线性复杂度的Vecchia推断在渐近意义上等价于具有立方复杂度的精确GP推断。这一结果同时为屏蔽效应提供了全新的通用结论。数值实验验证了上述发现，并表明Vecchia近似比协方差锥化、降秩近似等替代方法具有更高精度。