Causal Effect Identification from Multiple Incomplete Data Sources: A General Search-based Approach

A causal effect is defined as the distribution P (Y do(X), Z) where variables Y are observed, variables X are intervened upon (forced to values irrespective of their natural causes) and variables Z are conditioned on. Instead of placing various parametric restrictions based on background knowledge, we are interested in this paper in the question of identifiability: can the causal effect be uniquely determined from the distributions (data) we have and a graph representing our structural knowledge on the generating causal system. 1 In the most basic setting we are identifying causal effects from a single observational input distribution, corresponding to passively observed data. To solve such problems more generally than what is possible with the backdoor adjustment (Spirtes et al., 1993; Pearl, 2009; Greenland et al., 1999), Pearl (1995) introduced do-calculus, a set of three rules that together with probability theory enable the manipulation of interventional distributions. Shpitser and Pearl (2006a) and Huang and Valtorta (2006) showed that do-calculus is complete by presenting polynomial-time algorithms whose each step can be seen as a rule of do-calculus or as an operation based on basic probability theory. The algorithms have a high practical value because the rules of do-calculus do not by themselves provide an indication on the order in which they should be applied.