Well, you’re right. I’ll put a retraction in the post. Thank you!
By asking for citations you’re giving me too much credit—I’m playing the game a few levels below you. I was going off the “obvious” idea that if our prior gives nonzero probability to a hypothesis, and we feed the prior a sequence of bits that matches the hypothesis, then the posterior probability of the hypothesis should rise to 1. Took me some time to realize that’s not true at all!
Here’s a simple counterexample. Say our hypothesis is “all even bits are 0”, and our prior is an equal mixture of “all even bits are 0 and all odd bits are random” and “all odd bits are 0 and all even bits are random”. Note that our hypothesis starts out with probability 1⁄2 according to the prior. But if we feed the prior a sequence of all 0′s, which matches the hypothesis, the posterior probability of the hypothesis won’t go to 1. It will keep oscillating forever. The prior just happens to be too good at guessing this particular sequence without help from our hypothesis.
I’d be surprised if something like that happened with computable hypotheses and the universal prior, but I don’t have a proof and couldn’t find one in a few hours. So thanks again.
Well, you’re right. I’ll put a retraction in the post. Thank you!
By asking for citations you’re giving me too much credit—I’m playing the game a few levels below you. I was going off the “obvious” idea that if our prior gives nonzero probability to a hypothesis, and we feed the prior a sequence of bits that matches the hypothesis, then the posterior probability of the hypothesis should rise to 1. Took me some time to realize that’s not true at all!
Here’s a simple counterexample. Say our hypothesis is “all even bits are 0”, and our prior is an equal mixture of “all even bits are 0 and all odd bits are random” and “all odd bits are 0 and all even bits are random”. Note that our hypothesis starts out with probability 1⁄2 according to the prior. But if we feed the prior a sequence of all 0′s, which matches the hypothesis, the posterior probability of the hypothesis won’t go to 1. It will keep oscillating forever. The prior just happens to be too good at guessing this particular sequence without help from our hypothesis.
I’d be surprised if something like that happened with computable hypotheses and the universal prior, but I don’t have a proof and couldn’t find one in a few hours. So thanks again.