# Coincidences are Improbable

post by Mark Xu (mark-xu) · 2021-02-24T09:14:11.918Z · LW · GW · 13 comments

This is a link post for https://markxu.com/coincidences

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.

comment by AllAmericanBreakfast · 2021-02-24T20:12:05.561Z · LW(p) · GW(p)

I vote we abandon correlation does not imply causation in favor of connection does not imply direction. Or even better:

### connection alone, direction unknown

But I'd like it best if we had a positive version.

### Or was it their coincidence that caused it to be done?

Replies from: mark-xu, Raemon
comment by Mark Xu (mark-xu) · 2021-02-25T00:48:50.346Z · LW(p) · GW(p)

This is brilliant.

comment by Raemon · 2021-02-25T01:53:07.658Z · LW(p) · GW(p)

This is great. (I feel like the scansion could be improved slightly, but it'd be great as top level post)

But, reminds me that there's one more option here. Sometimes, instead of A causing B, or C causing A-and-B, A prevents B. This is similar (in some sense identical to "B causing A", but it prompts a different series of followup questions.

i.e. When crime happens, cops show up. When a disease happens, white blood cells show up. This isn't (quite) because the crime causes cops or disease causes white-blood cells. B does cause A, here, but, I think asking "is A's proximate 'goal' to prevent B" is a useful question to ask and check for, to help find models of what's going on.

Replies from: AllAmericanBreakfast
comment by AllAmericanBreakfast · 2021-02-25T03:10:05.509Z · LW(p) · GW(p)

Thanks for the push, I think the scansion is better optimized now.

Replies from: Raemon
comment by Raemon · 2021-02-25T03:22:06.693Z · LW(p) · GW(p)

### Did one cause the other? Did the other cause the one?

I'm not 100% sure I grok the meter, but I think this lines reads better in the new scansion if it's "Did one thing cause the other? Or the other cause the one?"

(fake edit: oh, I parsed the syllable emphasis wrong the first thing. For some reason it felt like "did" wasn't the downbeat syllable, so I started counting/feeling at "one")

comment by Measure · 2021-02-24T16:55:28.229Z · LW(p) · GW(p)

Occasionally I hear a song on the radio that I don't remember hearing in a while but that I had been thinking about only a day or two before. This surprised me, so I started thinking of possible explanations. My first thought was that I think about a lot of things and I hear a lot of songs on the radio so maybe there was bound to be a connection somewhere via the birthday paradox and that this was just more salient than the many times I had heard an unrelated song. I started making a mental note every time it happened, and even after considering the above, it still felt like it was happening too often to be pure coincidence. I currently think this happens because...

The station doesn't draw songs from the entire library of songs they play but rather selects from a smaller subset that gets updated every week or month. This song was recently added to the rotation and I already heard it once recently without remembering which prompted me to think about it.

comment by Daniel V · 2021-02-24T19:09:51.740Z · LW(p) · GW(p)

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).

Replies from: mark-xu
comment by Mark Xu (mark-xu) · 2021-02-24T19:43:19.545Z · LW(p) · GW(p)

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"

Replies from: philh, Daniel V
comment by philh · 2021-02-27T16:03:54.902Z · LW(p) · GW(p)

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.)

comment by Daniel V · 2021-02-24T19:48:03.349Z · LW(p) · GW(p)

Oh yeah, definitely agree!

comment by HDMI Cable · 2021-02-24T18:13:13.622Z · LW(p) · GW(p)

Some event  caused by both has been conditioned upon; new introductions have improbable attribute combinations because your friend seeks those combinations out.

This reads quite a bit like some sort of reverse-Barnum Effect, where instead of people trying to assign causality to vague descriptions / coincidences, they try to view linked (because of them) things as being coincidences.

comment by ForensicOceanography · 2021-02-25T17:53:45.067Z · LW(p) · GW(p)

Coincidences are improbable, but improbable thing happens. Given any two events A and B, each one with probability 0.01/day, we expect that in about one day in 10000 both will happen. Seems pretty low, but what happens if there are several events A, B, C, ...., Z, each one with probability 0.01?

If we consider 25 distinct improbable events, there are 25X24/2 = 300 pair of events, and each of those pair has a probability of 10^(-4) of happening in a day. Therefore, in about 3% of days (that is, one a month) you will observe a coincidence, which is not anymore that low.

In some of your examples, you are considering things that must have a well-defined cause (someone must have damaged your couch; something must be causing your skin irritation), so if the only thing new is the other event it is reasonable to guess a causal link (but what if you have a new dog and a new cat? Or if you changed lotion and you are also allergic to your new dog?). But in real life it is usually very difficult to insulate only one possible cause.

The latitude of the Great Pyramid of Giza coincides with the value of the speed of light in m/s, and the pyramids are roughly aligned with theposition of the Orion's Belt constellation in 12500 BC, and both these things are very unlikely, but I do not think there is something to uncover.

comment by Idan Arye · 2021-02-24T19:53:57.048Z · LW(p) · GW(p)

Coincidences can be evidence for correlation and therefore evidence for causation, as long as one remembers that evidence - like more things than most people feel comfortable with - are quantitative, not qualitative. A single coincidence, of even multiple coincidences, can make a causation less improbable - but it can still be considered very improbable until we get much more evidence.