Hierarchical Decompositions for the Computation of High-Dimensional Multivariate Normal Probabilities

We present a hierarchical decomposition scheme for computing the $\textit{n}$-dimensional integral of multivariate normal probabilities that appear frequently in statistics. The scheme exploits the fact that the formally dense covariance matrix can be approximated by a matrix with a hierarchical low-rank structure. It allows the reduction of the computational complexity per Monte Carlo sample from O(n2) to O(mn+knlog(n/m)), where $\textit{k}$ is the numerical rank of off-diagonal matrix blocks and $\textit{m}$ is the size of small diagonal blocks in the matrix that are not well-approximated by low-rank factorizations and treated as dense submatrices. This hierarchical decomposition leads to substantial efficiencies in multivariate normal probability computations and allows integrations in thousands of dimensions to be practical on modern workstations. Supplementary material for this article is available online.