Equations defining probability tree models

TL;DR

This paper explores the algebraic structure of coloured probability tree models, revealing how their invariants relate to odds ratios and providing conditions for these models to form toric varieties within probability simplices.

Contribution

It characterizes the algebraic invariants of coloured probability tree models and links their structure to toric varieties, enhancing understanding of their geometric properties.

Findings

Generators of the ideal are read from the tree graph as odds ratio differences.

Tree models' sum-to-one conditions translate into algebraic constraints on probabilities.

Necessary and sufficient conditions for staged trees to be toric varieties are identified.

Abstract

Coloured probability tree models are statistical models coding conditional independence between events depicted in a tree graph. They are more general than the very important class of context-specific Bayesian networks. In this paper, we study the algebraic properties of their ideal of model invariants. The generators of this ideal can be easily read from the tree graph and have a straightforward interpretation in terms of the underlying model: they are differences of odds ratios coming from conditional probabilities. One of the key findings in this analysis is that the tree is a convenient tool for understanding the exact algebraic way in which the sum-to-1 conditions on the parameter space translate into the sum-to-one conditions on the joint probabilities of the statistical model. This enables us to identify necessary and sufficient graphical conditions for a staged tree model to be a toric variety intersected with a probability simplex.