Ada Palmer:
events which are improbable and proximal are likely to have a causal link
I usually feel fine after eating food. One day, I decided to try a new dish at a restaurant. Afterward, my stomach is upset. I suspect that the new dish caused my stomachache. How justified is this suspicion?
Suppose events and both have a probability of occurring, and you observe both. This event favors various hypotheses over each other to the extent that they sharply predicted . A hypothesis that has assigns times more probability mass to than hypotheses that suppose and are independent.
More concretely, a hypothesis that postulates a strong causal link between and might have . This hypothesis is favored over a hypothesis that has . More generally, if you observe two improbable things, this is evidence that the presence of one observation makes the other more likely, with the evidence getting stronger as the connection between the two events strengthens.
Coincidences happen, but they are improbable. If you get a dog and your couch starts getting damaged, your dog is probably doing it. If your skin gets irritated and you recently switched lotion brands, you're probably allergic to the new brand. If my friend and I both saw someone six feet tall with red hair, we probably saw the same person. If your friend introduces you to someone that is both vegan and plays Magic the Gathering, you probably forget that your friend is also vegan and plays Magic the Gathering.
There are four ways events can be causally linked, only two of which are direct:
- causes ; your dog caused the couch damage.
- causes ; your skin irritation is caused by the new lotion brand.
- Some event causes both; the same 6-foot person causes both you and your friend to see them.
- Some event caused by both has been conditioned upon; new introductions have improbable attribute combinations because your friend seeks those combinations out.
When enough coincidences happen, start looking for a causal 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).
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.)