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

Computation of Gaussian orthant probabilities in high dimension