Staged tree models with toric structure

TL;DR

This paper explores the algebraic and geometric properties of staged tree models, revealing that many have a toric structure when viewed through appropriate coordinate transformations, which simplifies their analysis.

Contribution

It demonstrates that the class of staged tree models with toric structure extends beyond balanced trees via coordinate changes, advancing understanding of their algebraic properties.

Findings

Many staged tree models possess a toric structure after coordinate transformation.

Balanced trees are a subset where the model is the intersection of a toric variety and a probability simplex.

Open problem remains whether all staged tree models are toric in some coordinate system.

Abstract

A staged tree model is a discrete statistical model encoding relationships between events. These models are realised by directed trees with coloured vertices. In algebro-geometric terms, the model consists of points inside a toric variety. For certain trees, called balanced, the model is in fact the intersection of the toric variety and the probability simplex. This gives the model a straightforward description, and has computational advantages. In this paper we show that the class of staged tree models with a toric structure extends far outside of the balanced case, if we allow a change of coordinates. It is an open problem whether all staged tree models have toric structure.