Coincidences are Improbable

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 \(A\) and \(B\) both have a probability \(0.01\) of occurring, and you observe both. This event favors various hypotheses over each other to the extent that they sharply predicted \(A \land B\). A hypothesis that has \(P(B \mid A) = c * 0.01\) assigns \(c\) times more probability mass to \(A \land B\) than hypotheses that suppose \(A\) and \(B\) are independent.

More concretely, a hypothesis that postulates a strong causal link between \(A\) and \(B\) might have \(P(B \mid A) = 0.9\). This hypothesis is favored \(90:1\) over a hypothesis that has \(P(B \mid A) = P(B) = 0.01\). 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:

• \(A\) causes \(B\); your dog caused the couch damage.
• \(B\) causes \(A\); your skin irritation is caused by the new lotion brand.
• Some event \(C\) causes both; the same 6-foot person causes both you and your friend to see them.
• Some event \(D\) 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.