# Estimating orthant probabilities of high dimensional Gaussian vectors with an application to set estimation

arXiv: https://arxiv.org/abs/1603.05031 · HTML: https://mlsanity.com/paper/1603.05031/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Mar 2016.

## Built on by (most cited first, 3 of 3)

- [Properties of Discrete Sliced Wasserstein Losses](https://mlsanity.com/paper/2307.10352/) (2023)
- [Fixing the NTK: From Neural Network Linearizations to Exact Convex Programs](https://mlsanity.com/paper/2309.15096/) (2023)
- [ReLU Networks as Random Functions: Their Distribution in Probability Space](https://mlsanity.com/paper/2503.22082/) (2025)

## Abstract (arXiv)

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.