Causal inference via algebraic geometry: necessary and sufficient conditions for the feasibility of discrete causal models

We provide a scheme for inferring causal relations from uncontrolled statistical data which makes use of all of the information in the joint probability distribution over the observed variables rather than just the conditional independence relations. We focus on causal models containing just two observed variables, each of which is binary. We allow any number of latent variables and we do not impose any restriction on the manner in which the observed variables may depend functionally on the latent ones. In particular, the noise need not be additive. We provide an inductive scheme for classifying causal models into distinct observational equivalence classes. For each observational equivalence class, we provide a procedure for deriving, using techniques from algebraic geometry, necessary and sufficient conditions on the joint distribution for the feasibility of the class. Connections and applications of these results to the emerging field of quantum causal models are also discussed.