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IlyaShpitser comments on Causal Diagrams and Causal Models - Less Wrong
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Sorry, what you are missing is T and Y could be confounded by unobserved variables. That is, the real graph is:
z -> T -> Y, with T <- U -> Y, with U unobserved. Then if you control for T, you will get an open path z -> T <- U -> Y which is not causal. In general if your graph is
T -> Y <- U -> T, the causal effect is not a functional of the observed data. However with some parametric assumptions you can obtain the causal effect as a functional of the observed data if there is an instrument z.
Oh... so the idea in your second paragraph is that when you hold T constant, a change in z suggests an equal and opposite change in U (measuring by their mean effect on T). Then that change affects Y.
That's exactly right. The fact that for treatment T, and outcome Y, there is generally an unobserved common cause U of T and Y is in some sense the fundamental problem of causal inference. The way out is either:
(a) Make parametric assumptions and find instrumental variables (econometrics, mendelian randomization)
(b) Try to observe U (epidemiology, etc.)
(c) Randomize T (statistics, empirical science)
There are some other lesser known ways as well:
(d) Find an unconfounded mediator W that intercepts all causal influence from T to Y:
T -> W -> Y
Then use the "front-door criterion."