Graphoids Over Counterfactuals

Augmenting the graphoid axioms with three additional rules enables us to handle independencies among observed as well as counterfactual variables. The augmented set of axioms facilitates the derivation of testable implications and ignorability conditions whenever modeling assumptions are articulated in the language of counterfactuals. 1 Motivation Consider the causal Markov chain X → Y → Z which represents the structural equations: y = f(x, u1) (1) z = g(y, u2) (2) with u1 and u2 being omitted factors such that X, u1, u2 are mutually independent. It is well known that, regardless of the functions f and g, this model implies the conditional independence of X and Z given Y , written X ⊥⊥ Z | Y (3) This can be readily derived from the independence of X, u1, and u2, and it also follows from the d-separation criterion, since Y blocks all paths between X to Z. However, the causal chain can also be encoded in the language of counterfactuals by writing: Yx(u) = f(x, u1) (4) Zxy(u) = g(y, u2) = Zy(u) (5) where u stands for all omitted factors (in our case u = {u1, u2}) and Yx(u) stands for the value that Y would take in unit u had X been x. Accordingly, the functional and independence assumptions embedded in the chain model translate into the following counterfacutal 1 To appear in Journal of Causal Inference. TECHNICAL REPORT R-396 August 2014