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

TL;DR

This paper introduces a novel two-step method combining deterministic quadrature and stochastic Monte Carlo techniques, including an innovative asymmetric nested Monte Carlo algorithm, to efficiently estimate high-dimensional Gaussian orthant probabilities and apply them to set estimation.

Contribution

It presents a new high-dimensional probability estimation approach that combines deterministic and stochastic methods with a novel anMC algorithm, improving efficiency over existing techniques.

Findings

The method outperforms state-of-the-art techniques in numerical experiments.

The asymmetric nested Monte Carlo algorithm enhances efficiency in the remainder estimation.

Application to Gaussian random fields enables conservative set estimation without Markov assumptions.

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.