Algebraic geometry of discrete interventional models

TL;DR

This paper explores the algebraic and geometric structures of discrete interventional models, introducing a new framework for soft interventions in staged tree models and analyzing their algebraic properties.

Contribution

It develops a formalism for modeling soft interventions in staged tree models and provides criteria to identify when these models are toric varieties, extending understanding of their algebraic structure.

Findings

Derived combinatorial criteria for toric ideals in interventional staged tree models

Established conditions under which discrete interventional DAG models are toric varieties

Provided methods to find defining equations of these models

Abstract

We investigate the algebra and geometry of general interventions in discrete DAG models. To this end, we introduce a theory for modeling soft interventions in the more general family of staged tree models and develop the formalism to study these models as parametrized subvarieties of a product of probability simplices. We then consider the problem of finding their defining equations, and we derive a combinatorial criterion for identifying interventional staged tree models for which the defining ideal is toric. We apply these results to the class of discrete interventional DAG models and establish a criteria to determine when these models are toric varieties.