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.

在非负整数值多元无限可分分布理论中，多元泊松分布起着关键作用。与单变量情况一样，任何非负整数值的无限可分多元分布都可以通过属于复合泊松族的多元分布来近似。多元泊松分布是该家族的重要成员。近年来，多元泊松分布也获得了实际重要性，因为它们用作描述具有正协方差结构的计数数据的模型。然而，由于计算多元泊松概率质量函数 (pmf) 及其相应的累积分布函数 (cdf) 的计算复杂性，它们在这些计数模型中的使用受到限制。由于多元泊松概率分布的大部分理论属性似乎已经为人所知，因此本文的主要重点是提出更有效的算法来计算该 pmf。利用泊松多元分布随机向量的众所周知的性质，我们在本文中提出了一种基于找到线性丢番图方程组的所有解来计算该 pmf 的直接方法。这种新方法根据 PMF 现有的递归关系的使用补充了现有的程序。我们将我们的新方法与应用于一组略有不同的递归关系的现有方法进行比较，这些方法更容易评估。还给出了这组新的递归关系的证明。因此，提出了几种算法，其中一些算法基于新方法，一些算法使用递归关系。为了测试这些算法，我们在计算部分提供了广泛的分析。基于本节中的实验，我们得出结论，寻找一组线性丢番图方程的所有解的方法在计算上比使用递推关系来评估多元泊松分布随机向量的 pmf 的方法更有效。