What do we know about counterfactuals in linear models?
Here is a neat result concerning the testability of counterfactuals in linear systems.
We know that counterfactual queries of the form P(Yx=y|e) may or may not be empirically identifiable, even in experimental studies. For example, the probability of causation, P(Yx=y|x',y') is in general not identifiable from experimental data (Causality, p. 290, Corollary 9.2.12) when X and Y are binary.1 (Footnote-1: A complete graphical criterion for distinguishing testable from nontestable counterfactuals is given in Shpitser and Pearl (2007, upcoming)).
This note shows that things are much friendlier in linear analysis:
Claim A. Any counterfactual query of the form E(Yx |e) is empirically identifiable in linear causal models, with e an arbitrary evidence.
Claim B. E(Yx|e) is given by
E(Yx|e) = E(Y|e) + T [x – E(X|e)] (1)
where T is the total effect coefficient of X on Y, i.e.,
T = d E[Yx]/dx = E(Y|do(x+1)) – E(Y|do(x)) (2)
Thus, whenever the causal effect T is identified, E(Yx|e) = is identified as well.
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The definition of the back-door condition (Causality, page 79, Definition 3.3.1) seems to be contrived. The exclusion of descendants of X (Condition (i)) seems to be introduced as an after fact, just because we get into trouble if we dont. Why cant we get it from first principles; first define sufficiency of Z in terms of the goal of removing bias and, then, show that, to achieve this goal, you neither want nor need descendants of X in Z.