One often hears it said that “absence of evidence is not evidence of absence”. For example, if you excavate some fossil sauropods and they don’t have preserved feathers, that not evidence that sauropods didn’t have feathers.

Oh yes it is.

This is an example of a mantra that’s short, catchy, and wrong. Every time you see absence of evidence, you are accumulating evidence of absence. What you’re not doing is proving absence — not with any single observation, at least.

Here’s an example. You toss a coin, and it comes down heads. You now have evidence of the presence of a head side (hurrah!) but only absence of evidence for a tails side. It’s perfectly likely that the coin has a tails side but merely happened to come up heads. So the absence of evidence for a tails side is certainly not proof that there is not a tails side.

But now suppose you toss the coin ten times, and it comes up heads every time. For a fair coin, there is less than one chance in a thousand of that happening (2^10 = 1024). Now let’s toss it 20 times. If you still see heads every time, you’re seeing a one-in-a-million event (2^20 = 1048576), and if you’re still seeing only heads after 100 tosses, that is a 1-in-1267650600228229401496703205376 event.

By this point, I’m hoping you’d be concluding with reasonable certainly that you had a two-headed coin. What’s happening is that each observation of absence (no tails side) is increasingly your confidence that there is no such side.

Absence of evidence is evidence of absence.