Algebraic geometry of discrete interventional 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 parameterized subvarieties of a product of probability simplices. We then consider the problem of finding their defining equations, and we derive sufficient combinatorial conditions on an interventional staged tree model to have a defining ideal that is toric. We apply these results to the class of discrete interventional DAG models and establish sufficient graphical conditions to determine when these models are toric varieties.