Holonomic Gradient Method for Two Way Contingency Tables

The holonomic gradient method gives an algorithm to efficiently and accurately evaluate normalizing constants and their derivatives. We apply the holonomic gradient method in the case of the conditional Poisson or multinomial distribution on two way contingency tables. We utilize the modular method in computer algebra for an efficient and exact evaluation, and we discuss on complexities of these algorithms and their implementation. We also discuss on a theoretical aspect of the distribution from the viewpoint of the conditional maximum likelihood estimation.