Restrictions of Pfaffian system and hypergeometric system of contingency table

Exponential families on a finite set are well-studied in statistics, but they are still a source of many problems. For example, marginal likelihood integral in Bayesian inference for conjugate prior naturally gives rise to a hypergeometric integral whose exact formula/evaluation is unknown. Bayesian inference corresponds to contiguity relation, which is an automorphism of de Rham cohomology group defined by the integrand. When the exponential family is generic, the integral is a Gel'fand-Kapranov-Zelevinsky hypergeometric function and the contiguity structure is described in terms of Pfaffian system (system of 1st order linear P.D.E.'s). It is natural to ask the following question: can we restrict Pfaffian system of a generic exponential family to a non-generic one? In this talk, we propose a method to keep track of such restrictions using a technique of singular boundary value problem. In the latter half of the talk, we pay a special attention to two-way (incomplete) contingency table, a class of exponential family. Restriction method combined with the theory of hyperplane arrangement provides a new combinatorial description of the Pfaffian system. This Pfaffian system generalizes familiar hypergeometric systems such as pFp-1, Appell-Lauricella's FA,FB,FD, Tsuda and Aomoto Gel'fand system. The talk is partially based on an on-going joint work with Vsevolod Chestnov, Federico Gasparotto, Manoj K. Mandal, Pierpaolo Mastrolia, Henrik J. Munch and Nobuki Takayama.