Algebraic geometry of discrete interventional models
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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.
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