Computation of Gaussian orthant probabilities in high dimension

We study the computation of Gaussian orthant probabilities, i.e. the probability that a Gaussian variable falls inside a quadrant. The Geweke-Hajivassiliou-Keane (GHK) algorithm (Geweke, Comput Sci Stat 23:571-578 1991, Keane, Simulation estimation for panel data models with limited dependent variables, 1993, Hajivassiliou, J Econom 72:85-134, 1996, Genz, J Comput Graph Stat 1:141-149, 1992) 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. 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. We then generalize this idea to arbitrary covariance matrices using Sequential Monte Carlo 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.