Causal Inference on Data Containing Deterministic Relations

Data containing deterministic relations cannot be handled by current constraint-based causal learning algorithms; they entail conditional independencies that cannot be represented by a faithful graph. Violation of the faithfulness property is characterized by an information equivalence of two sets of variables with respect to a reference variable. The conditional independencies do not provide information about which set should be connected to the reference variable. We propose to use the complexity of the relationships as criterion to determine adjacency. Correct decisions are made under the assumption that the complexity of relations does not increase along a causal path. This paper defines an augmented Bayesian network which explicitly models deterministic relations. The faithfulness property is redefined by using a generalized definition of the d-separation criterion, which also gives the conditional independencies following from deterministic relations, and by limiting the conditional independencies that are graphically described with the simplicity condition. Based on this, an extension to the PC learning algorithm is developed that allows the construction of minimal augmented Bayesian networks from observational data. Correct models are learned from data generated by a set of structural equations.