Naftali Weinberger, Path-Specific Effects (Brit J Phil Sci 70 (2019) 53-76

Weinberger makes a very plausible case that interventionism—tendance Pearl (2001)—can explain path-dependent effects in a way that probability-raising accounts of causation cannot. 

The problem of path-specific causation is illustrated by the “Hesslow case”. One side effect of birth control pills is to increase the risk of thrombosis; apparently they alter blood chemistry in a way that makes clotting more likely. But they also decrease the risk of thrombosis!  They prevent pregnancy, which also has thrombosis as a side effect. The net effect of these two causal paths might be positive (so the pill increases the chance of thrombosis overall), negative, or even exactly zero.  This situation is obviously different from a simple risk-raising or lowering effect—that of statins, say—in ways that are practically relevant. Suppose that taking the pill lowers the risk of thrombosis overall. Telling an infertile person to take it for that reason would be bad advice. The challenge for probability-raising accounts of causation is to explain that difference. How can taking the pill both raise and lower the chances of thrombosis?

Probability theorists have attempted to address this problem, of course, and Weinberger rehearses the debate between Nancy Cartwright and Ellery Eells about how to do so. I’ll elide the details here, but Weinberger’s claim is that the resources probability theorists can appeal to—differences in background variables and (per Cartwright) singular or token causation—are not enough. By contrast, Pearl’s approach allows for a neat system of distinguishing betwen the paths.  

Which is this. To determine whether (and indeed how much) the pill contributes to reducing the risk of thrombosis via the blood chemistry path, you intervene (in the Woodwardian sense) on the mediator of the other path. That blocks that other causal path, allowing you to see the effect via the path you’re interested in.

More formally, you set the value of the “pregnancy” variable to “isn't", breaking its causal relation to taking the pill, and look at the difference in the value of the outcome “thrombosis” variable when “takes the pill” is set to first “does" and then “does not". (That difference can be calculated using the stuctural equations in your model.) In the Hesslow case, the difference will be non-zero, which tells you that there is an effect along that path.

Things get more complicated when the other path is direct, that is, has no mediator. To see that the pill influences thrombosis via pregnancy, you can’t intervene on a mediator in the other path, since (let’s stipulate) there isn’t one. (Or perhaps there is one, but it interacts with pregnancy, and so shouldn’t be intervened on.) The framework can handle this.  Rather than blocking the other path, you fix its contribution, by holding the original cause fixed.  Then you intervene on the mediator of the path you are interested in as if there was a change in the original cause.  Since the other path is fixed, this allows you to see the impact of the path youre interested in. So in Hesslow you hold fixed that the pill is taken, and then intervene on the pregnancy variable to give it the value it would have if the pill hadn’t been taken. You compare the outcome with what happens if the pregnanacy variable is allowed to take its “pill taken” value.  Again, the stuctural equations will tell you that the difference is non-zero, so there is causal influence down the pregnancy path.

Weinberger’s framework also usefully distinguishes between “necessary” and “sufficient” path-dependent effects. The former compares the effect along that path to the total effect; the latter compares the effect along that path to the result where the original cause doesn’t happen. These will provide answers to questions with different contrasts. If you took the pill and got thrombosis, but not pregnant, you probably want to know how the risk would have been different if you hadn’t taken the pill (the sufficient effect along the direct path). But you might also want to know what it would have been if the pill hadn't prevented pregnancy (the necessary effect along the indirect path).

The necessary/sufficient terminology looks counterintuitive, but it fits nicely with the point that the total effect is the sum of the sufficient direct effect and the necessary indirect effect, and vice versa. (This point is made much clearer by the diagrams in the paper.) This allows a decomposition of the total effect that avoids the pitfalls of treating causes as simply additive—even where there is interaction between mediators along the different paths. (Although Weinberger warns against slipping from the possibility of mathematical decomposition to the idea that there really is an ontologically independent contribution along each path.) 

In all this, of course, Weinberger is very much in the spirit of Pearl’s own criticism of attempts to understand causation in terms of probability, or any other sort of correlation. (See chapter 1 of the Book of Why for a very friendly discussion.) But the detailed application of Pearl’s framework to the philosophical discussion of the Hesslow case, and other path-dependent effect cases, is independently valuable. It took me a little while to puzzle out some of the logic—the discussion of Eells versus Cartwright in particular had me scribbling on paper. It’s really helpful, I think, to pay attention to the negation load you’re imposing on your reader. (I appreciate I am also guilty here.) But this is an excellent, and I think correct, example of how the interventionist framework can help with longstanding philosophical puzzles.