On computing the multivariate poisson probability distribution

Official URL: https://dx.doi.org/10.1007/s11009-023-10036-z

Within the theory of non-negative integer valued multivariate infinitely divisible distributions, the multivariate Poisson distribution plays a key role. As in the univariate case, any non-negative integer valued infinitely divisible multivariate distribution can be approximated by a multivariate distribution belonging to the compound Poisson family. The multivariate Poisson distribution is an important member of this family. In recent years, the multivariate Poisson distributions also has gained practical importance, since they serve as models to describe counting data having a positive covariance structure. However, due to the computational complexity of computing the multivariate Poisson probability mass function (pmf) and its corresponding cumulative distribution function (cdf), their use within these counting models is limited. Since most of the theoretical properties of the multivariate Poisson probability distribution seem already to be known, the main focus of this paper is on proposing more efficient algorithms to compute this pmf. Using a well known property of a Poisson multivariate distributed random vector, we propose in this paper a direct approach to calculate this pmf based on finding all solutions of a system of linear Diophantine equations. This new approach complements an already existing procedure depending on the use of recurrence relations existing for the pmf. We compare our new approach with this already existing approach applied to a slightly different set of recurrence relations which are easier to evaluate. A proof of this new set of recurrence relations is also given. As a result, several algorithms are proposed where some of them are based on the new approach and some use the recurrence relations. To test these algorithms, we provide an extensive analysis in the computational section. Based on the experiments in this section, we conclude that the approach finding all solutions of a set of linear Diophantine equations is computationally more efficient than the approach using the recurrence relations to evaluate the pmf of a multivariate Poisson distributed random vector.