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

Abstract

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.

References in corpus (2)

• Simulation of truncated normal variables
• Spatio-temporal exceedance locations and confidence regions

Cited by in corpus (7)

• Adaptive Design of Experiments for Conservative Estimation of Excursion Sets
• Integrals over Gaussians under Linear Domain Constraints
• Exploiting Low Rank Covariance Structures for Computing High-Dimensional Normal and Student-$t$ Probabilities
• Adaptive Batching for Gaussian Process Surrogates with Application in Noisy Level Set Estimation
• Properties of Discrete Sliced Wasserstein Losses
• O(N) Hierarchical algorithm for computing the expectations of truncated multi-variate normal distributions in N dimensions
• Referenced Thermodynamic Integration for Bayesian Model Selection: Application to COVID-19 Model Selection