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

2015/12/21 by Koyama, Tamio
#16S32 62H10 #33E20 #Computation (stat.CO) #FOS: Computer and information sciences #FOS: Mathematics #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.1512.06564

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.

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