2014/11/05 by James Ridgway, Ridgway, James
Computer Science · Mathematics · #Bayesian Methods and Mixture Models #Computation (stat.CO) #FOS: Computer and information sciences #Markov Chains and Monte Carlo Methods #Target Tracking and Data Fusion in Sensor Networks
paper · pdf · doi:10.48550/arxiv.1411.1314
openalex publication_date 2014/11/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.