I am studying Bayes‘ theorem in school right now, probably for the third or fourth time.

Bayes' formula, courtesy of Wikipedia

Nevertheless, like many aspects of mathematics, (and everything I suppose) I think I learn it just a little bit differently and better each time.

The big realization for me on this occasion is that the theorem is succinctly and practically encapsulated by Carl Sagan‘s famous epigram:

Extraordinary claims require extraordinary evidence.

Basically, we can assume that P(B|A) is high. B is the observed phenomenon, and A is the explanation, so it wouldn’t make much sense to talk about them together if A didn’t do a good job explaining why B occurs.

Then, it’s just a matter of the ratio between P(A) and P(B). If A is a wild, crazy, and new explanation, that it better have some wild crazy and new data to back it up. That will help keep P(A) / P(B) close to 1, and thus P(A|B) close to 1, meaning that A is a good theory.

Now obviously I’m not the first person to think this up (in fact, neither was Sagan). But it really clicked for me, and it’s fun to share when that happens.