Perturbation method for determining the group of invariance of hierarchical models

TL;DR

This paper introduces a perturbation method to identify the largest group of invariance in hierarchical models, simplifying previous proofs and applying to general contingency table models with generic factor levels.

Contribution

It presents a new perturbation approach that simplifies determining the invariance group of toric ideals in hierarchical models, extending previous work.

Findings

Determines the invariance group as a wreath product indexed by a poset.

Provides a simpler proof for the invariance group of hierarchical models.

Applies to models with generic factor levels.

Abstract

We propose a perturbation method for determining the (largest) group of invariance of a toric ideal defined in Aoki and Takemura [2008a]. In the perturbation method, we investigate how a generic element in the row space of the configuration defining a toric ideal is mapped by a permutation of the indeterminates. Compared to the proof in Aoki and Takemura [2008a] which was based on stabilizers of a subset of indeterminates, the perturbation method gives a much simpler proof of the group of invariance. In particular, we determine the group of invariance for a general hierarchical model of contingency tables in statistics, under the assumption that the numbers of the levels of the factors are generic. We prove that it is a wreath product indexed by a poset related to the intersection poset of the maximal interaction effects of the model.