On the ideals of equivariant tree models

TL;DR

This paper introduces equivariant tree models in algebraic statistics, demonstrating how their ideals can be derived from star models and showing that these ideals are generated by flattenings at vertices, advancing understanding of their algebraic structure.

Contribution

It proves that the ideals of general equivariant tree models can be fully determined from star models and their flattenings, providing a comprehensive algebraic characterization.

Findings

Ideals for general trees can be derived from star models.

The entire ideal is generated by flattenings at vertices.

The approach unifies various existing tree models.

Abstract

We introduce equivariant tree models in algebraic statistics, which unify and generalise existing tree models such as the general Markov model, the strand symmetric model, and group based models. We focus on the ideals of such models. We show how the ideals for general trees can be determined from the ideals for stars. The main novelty is our proof that this procedure yields the entire ideal, not just an ideal defining the model set-theoretically. A corollary of theoretical importance is that the ideal for a general tree is generated by the ideals of its flattenings at vertices.