Algebraic algorithm for direct sampling from toric models and hypergeometric functions

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.