Multivariate Gaussian cumulative distribution functions as the marginal likelihood of their dual Bayesian probit models

The computation of multivariate Gaussian cumulative distribution functions is a key step in many statistical procedures, often representing a crucial computational bottleneck. Over the past few decades, efficient algorithms have been proposed to address this problem, mainly using Monte Carlo solutions. This work highlights a connection between the multivariate Gaussian cumulative distribution function and the marginal likelihood of a tailored dual Bayesian probit model. Consequently, any method that approximates such a marginal likelihood can be used to estimate the quantity of interest. We focus on the approximation provided by the expectation propagation algorithm. Its empirical accuracy and polynomial computational cost make it an appealing choice, especially for tail probabilities, even if theoretical guarantees are currently limited. Its efficiency, accuracy and stability are shown for multiple correlation matrices and integration limits, highlighting a series of advantages over state-of-the-art alternatives.