Algebraic geometry of discrete interventional models

We investigate the algebra and geometry of general interventions in discrete DAG models. To this end, we develop the formalism to study these models as subvarieties of multiprojective space and introduce a theory for modeling soft interventions in the more general family of staged tree models. 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. This criterion, when combined with a new characterization of decomposable DAG models in terms of their associated staged trees, specializes to a graphical criterion in the case of discrete interventional DAG models.