Computation of Gaussian orthant probabilities in high dimension

TL;DR

This paper analyzes and improves the computation of high-dimensional Gaussian orthant probabilities by interpreting GHK as a sequential importance sampling estimator, proposing particle filter methods, and extending to mixtures and truncated Gaussians.

Contribution

It reinterprets GHK as a sequential importance sampling estimator, introduces particle filter improvements, and extends the framework to mixtures and truncated Gaussians.

Findings

GHK variance diverges exponentially with dimension for AR(1)

Particle filters significantly improve estimation accuracy

Framework extension to mixture and truncated Gaussians

Abstract

We study the computation of Gaussian orthant probabilities, i.e. the probability that a Gaussian falls inside a quadrant. The Geweke-Hajivassiliou-Keane (GHK) algorithm [Genz, 1992; Geweke, 1991; Hajivassiliou et al., 1996; Keane, 1993], is currently used for integrals of dimension greater than 10. In this paper we show that for Markovian covariances GHK can be interpreted as the estimator of the normalizing constant of a state space model using sequential importance sampling (SIS). We show for an AR(1) the variance of the GHK, properly normalized, diverges exponentially fast with the dimension. As an improvement we propose using a particle filter (PF). We then generalize this idea to arbitrary covariance matrices using Sequential Monte Carlo (SMC) with properly tailored MCMC moves. We show empirically that this can lead to drastic improvements on currently used algorithms. We also extend the framework to orthants of mixture of Gaussians (Student, Cauchy etc.), and to the simulation of truncated Gaussians.