Holonomic Gradient Method for Two Way Contingency Tables

Abstract

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 or some other tricks for an efficient and exact evaluation, and we compare them and discuss on their implementation. We also discuss on a theoretical aspect of the distribution from the viewpoint of the conditional maximum likelihood estimation. We decompose parameters of interest and nuisance parameters in terms of sigma algebras for general two way contingency tables with arbitrary zero cell patterns.

Intersection numbers of twisted cycles and cocycles for degenerate arrangements

“…As in [5] and [9], the contiguity relations are utilized for algebraic statistics. The hypergeometric polynomial is regarded as the normalizing constant of the hypergeometric distribution of two-way contingency tables with fixed marginal sums.…”

“…These bases are then used to investigate the contiguity relations in Section 5. As discussed in [9], explicit expressions for the contiguity relations are useful in algebraic statistics. In [5], we presented such expressions for the case of a general position in terms of the intersection numbers of twisted cohomology groups.…”