  The computation of Gaussian orthant probabilities has been extensively
studied for low-dimensional vectors. Here, we focus on the high-dimensional
case and we present a two-step procedure relying on both deterministic and
stochastic techniques. The proposed estimator relies indeed on splitting the
probability into a low-dimensional term and a remainder. While the
low-dimensional probability can be estimated by fast and accurate quadrature,
the remainder requires Monte Carlo sampling. We further refine the estimation
by using a novel asymmetric nested Monte Carlo (anMC) algorithm for the
remainder and we highlight cases where this approximation brings substantial
efficiency gains. The proposed methods are compared against state-of-the-art
techniques in a numerical study, which also calls attention to the advantages
and drawbacks of the procedure. Finally, the proposed method is applied to
derive conservative estimates of excursion sets of expensive to evaluate
deterministic functions under a Gaussian random field prior, without requiring
a Markov assumption. Supplementary material for this article is available
online.
