On the Intersection Property of Conditional Independence and its Application to Causal Discovery

Inferring causal relationships is a major challenge in science. In the last decades considerable effort has been made in order to learn causal statements from observational data. Causal discovery methods make assumptions that relate the joint distribution with properties of the causal graph. Constraintbased or independence-based methods [Pearl, 2009, Spirtes et al., 2000] and some score-based methods [Chickering, 2002, Heckerman et al., 1999] assume the Markov condition and faithfulness. A distribution is said to be Markov with respect to a directed acyclic graph (DAG) G if each d-separation in the graph implies the corresponding (conditional) independence; the distribution is faithful with respect to G if the reverse statement holds. These 1 two assumptions render the Markov equivalence class of the correct graph identifiable from the joint distribution, i.e. the skeleton and the v-structures of the graph can be inferred from the joint distribution [Verma and Pearl, 1991]. Methods like LiNGAM [Shimizu et al., 2006] or additive noise models [Hoyer et al., 2009, Peters et al., 2013] assume the Markov condition, too, but do not require faithfulness; instead, these methods assume that the structural equations come from a restricted model class (e.g.