The two “direct” causal links are the only ones we would really call “causal” regarding A and B.
But I am a big fan of “correlation implies causation.” It might not be between A and B specifically, but it means we’ve been able to detect something happening.
Sometimes even non-effects, when theory is strong enough, can indicate causation (though then the usual course of action is to control one of the paths to get an effect that you can talk about and publish). For example, you are about to eat an allergen, which you know causes side effects for you with p=1. You take Benadryl beforehand and have no side effects. There is no “effect” there (post state = pre state), but you can feel pretty sure Benadryl had a suppressing action on the allergen’s effects (and then you would follow-up with experiments where you ate the allergen without Benadryl or took the Benadryl without eating the allergen to see the positive and negative effects separately).
Seems worth mentioning that the four ways you list for events to be causally linked are the building blocks of d-separation, not the whole thing. E.g. “A causes X, X causes B” is a causal link, but not direct. And “A causes X, B causes X, X causes Y, and we’ve observed Y” is one as well. Or even: “A causes X, Y causes X, Y causes B, X causes Z, and we’ve observed Z”. (That’s the link between s and y in example 3 from your link.)
The two “direct” causal links are the only ones we would really call “causal” regarding A and B.
But I am a big fan of “correlation implies causation.” It might not be between A and B specifically, but it means we’ve been able to detect something happening.
Sometimes even non-effects, when theory is strong enough, can indicate causation (though then the usual course of action is to control one of the paths to get an effect that you can talk about and publish). For example, you are about to eat an allergen, which you know causes side effects for you with p=1. You take Benadryl beforehand and have no side effects. There is no “effect” there (post state = pre state), but you can feel pretty sure Benadryl had a suppressing action on the allergen’s effects (and then you would follow-up with experiments where you ate the allergen without Benadryl or took the Benadryl without eating the allergen to see the positive and negative effects separately).
I am using the word “causal” to mean d-connected, which means not d-seperated. I prefer the term “directly causal” to mean A->B or B->A.
In the case of non-effects, the improbable events are “taking Benadryl” and “not reacting after consuming an allergy”
Seems worth mentioning that the four ways you list for events to be causally linked are the building blocks of d-separation, not the whole thing. E.g. “A causes X, X causes B” is a causal link, but not direct. And “A causes X, B causes X, X causes Y, and we’ve observed Y” is one as well. Or even: “A causes X, Y causes X, Y causes B, X causes Z, and we’ve observed Z”. (That’s the link between s and y in example 3 from your link.)
Oh yeah, definitely agree!