  Gaussian processes (GPs) are commonly used as models for functions, time
series, and spatial fields, but they are computationally infeasible for large
datasets. Focusing on the typical setting of modeling data as a GP plus an
additive noise term, we propose a generalization of the Vecchia (1988) approach
as a framework for GP approximations. We show that our general Vecchia approach
contains many popular existing GP approximations as special cases, allowing for
comparisons among the different methods within a unified framework.
Representing the models by directed acyclic graphs, we determine the sparsity
of the matrices necessary for inference, which leads to new insights regarding
the computational properties. Based on these results, we propose a novel sparse
general Vecchia approximation, which ensures computational feasibility for
large spatial datasets but can lead to considerable improvements in
approximation accuracy over Vecchia's original approach. We provide several
theoretical results and conduct numerical comparisons. We conclude with
guidelines for the use of Vecchia approximations in spatial statistics.
