Holonomic gradient method for the probability content of a simplex region with a multivariate normal distribution

TL;DR

This paper introduces a holonomic gradient method to efficiently compute the probability content of a simplex region under a multivariate normal distribution, extending inclusion-exclusion identities to faces of polyhedra.

Contribution

The authors generalize the inclusion-exclusion identity for polyhedra to their faces, enabling derivative calculations of probability content functions for polyhedral regions.

Findings

The method accurately computes probability content of simplex regions.

Derivatives of probability functions can be expressed as integrals over polyhedral faces.

The approach extends to general polyhedra in statistical computations.

Abstract

We use the holonomic gradient method to evaluate the probability content of a simplex region under a multivariate normal distribution. This probability equals to the integral of the probability density function of the multivariate Gaussian distribution on the simplex region. For this purpose, we generalize the inclusion--exclusion identity which was given for polyhedra, to the faces of a polyhedron. This extended inclusion--exclusion identity enables us to calculate the derivatives of the function associated with the probability content of a polyhedron in general position. We show that these derivatives can be written as integrals of the faces of the polyhedron.