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

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.