We show that Pfaffians or contiguity relations of hypergeometric functions of several variables give a direct sampling algorithm from toric models in statistics, which is a Markov chain on a lattice generated by a matrix $A$. A correspondence among graphical models and $A$-hypergeometric system is discussed and we give a sum formula of special values of $A$-hypergeometric polynomials. Some hypergeometric series which are interesting in view of statistics are presented.

翻译：我们证明了多变量超几何函数的Pfaffians或邻接关系为统计学中的环面模型提供了一种直接采样算法，该算法是在由矩阵$A$生成的格上的马尔可夫链。本文讨论了图模型与$A$-超几何系统之间的对应关系，并给出了$A$-超几何多项式特殊值的求和公式。同时，我们展示了一些在统计学视角下具有研究价值的超几何级数。