  Spatial statistics often involves Cholesky decomposition of covariance
matrices. To ensure scalability to high dimensions, several recent
approximations have assumed a sparse Cholesky factor of the precision matrix.
We propose a hierarchical Vecchia approximation, whose conditional-independence
assumptions imply sparsity in the Cholesky factors of both the precision and
the covariance matrix. This remarkable property is crucial for applications to
high-dimensional spatio-temporal filtering. We present a fast and simple
algorithm to compute our hierarchical Vecchia approximation, and we provide
extensions to non-linear data assimilation with non-Gaussian data based on the
Laplace approximation. In several numerical comparisons, including a filtering
analysis of satellite data, our methods strongly outperformed alternative
approaches.
