Thread: Philosophers have been obsessing over multi-path causes for over 40 years, but without @yudapearl's non-parametric #mediation techniques they have been unable to provide an account of #path-specific effects. Why not?

First, both philosophers and social scientists have focused on linear non-additive causal models. But in these models direct and indirect effects have properties that don't generalize to other cases.
In linear models, the total effect is the sum of the direct and indirect effect. One gets the direct effect by conditioning on the mediator, and the value at which you condition on the mediator doesn't matter. The indirect effect is the total effect minus the direct effect.
This makes it seem like the total effect can be decomposed into two independent contributions corresponding to the two paths. But when the treatment and mediator interact such a decomposition no longer makes sense. e.g. The direct effect depends on the value of the mediator.
So we can no longer unambiguously ask about the "contributions" of the direct or indirect paths, since there are no isolated contributions. If the paths do not decompose, how does it make sense to distinguish between the direct and indirect effects?
To answer this, you need the counterfactuals embedded in structural equations. The direct effect (DE) is the effect that a change in the treatment would have on the outcome were the mediator to have the same value that it would have had were the treatment not changed.
Let's unpack this. First, path-specific effects are relative to some change in the treatment from one value to another. Let's say from X=0 to X=1.
For the "natural" direct effect, we change the treatment from X=0 to X=1, while setting the mediator to the value it would have taken on were X=0. (For populations, this value can be different for different individuals, and must be set accordingly)
More generally, a path-specific effect of changing X from x to x' is the effect of changing X in this way while allowing the change to be "transmitted" via only one of the paths.
We've seen how to do this for DE. How do we do it for the indirect effect IE? (outside of the linear additive case, the IE is NOT the product of the path coefficients)
For the IE of changing X from 0 to 1, you hold it fixed at 0 and then intervene on the mediator to behave AS IF it were responding to a change in the treatment. That is you compare the values the mediator would take on when X=0 and X=1.
Contrary to what some have said, you do NOT need to include an additional variable along the direct path and then intervene on it in order to measure the IE. IE would be defined even if the treatment influenced the outcome at a distance (I.e. there were no omitted mediators).
Clearly, to give an account of path-specific effects, you need a formalism for representing counterfactuals. This is the second reason why philosophers were unable to give an account of path-specific effects.
Consider theorists of "probabilistic causality" in the 70's-90's. They had a basic, if non-general, account of what variables to condition on to eliminate confounding. They could describe the causal relationships that obtain in populations with different background factors.
So in the standard case where birth control influences thrombosis via pregnancy (and some direct path), they could talk about the net effect of BC on thrombosis in the whole population, and also in the population for whom BC does not influence pregnancy
But these were two totally different populations. The latter population does not give the DE for the former population. It is just a population with a different set of background factors.
Path-specific effects relate to the total effect only because they correspond to what the net effect WOULD be for the population were the treatment to influence the outcome in that population, but only via one of the paths.
The populations (or individuals) in which the total effect and the path-specific effects are measured do NOT have the same background factors, but still resemble one another in being governed by the same structural equations.
This is why it is important that in measuring DE, one does not intervene to set the mediator to just any value, but to the value it would have taken on had there not been the change in the value of X. This is given by the structural equation (or the #potentialoutcomes).
So without non-additive models, you can't see the subtleties in defining path-specific effects. Without counterfactuals, you can't give a proper account of them.
The closest any philosopher came to defining path-specific effects what Chris Hitchcock in "Tale of Two effects". He distinguishes between component effects and net effects and the former sometimes match DE.
But his component effects are not path-specific effects. His point is that we sometimes care about net effects and sometimes about component effects. But these are two independent causal concepts with no relation to one another.
But path-specific effects are what the net effect would be in a scenario where a change in the treatment were not transmitted via one of the paths. It is this connection that makes path-specific effects relate to the total effect, and not be some totally independent effect.
So it is only with Pearl's 2001 "Direct and Indirect Effects" that we get a general and correct account of path-specific effects, and the consequences of this are still underappreciated by both philosophers and scientists.
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