Alice correctly predicting the sequence that she achieves is evidence that causes a substantial update on our distribution over world-models, even if the two sequences are assigned equal probability in our distribution over sequences given that the null hypothesis is true.
Except that we're not updating all distributions of all possible world-models, or every single sequence would be equally surprising. You're implicitly looking for evidence that, say, Alice is clairvoyant - you've elevated that hypothesis to your awareness before you ever looked at the evidence.
Except that we're not updating all distributions of all possible world-models, or every single sequence would be equally surprising.
If you don't even know what you mean by surprise (because that's what we're ostensibly trying to figure out, right?), then how can you use the math to deduce that some quantitative measure of surprise is equal in all cases?
I still think this is just a confusion over having a distribution over sequences of coin flips as opposed to a distribution over world-models.
Suppose you have a prior distribution over a space of hypothes...
Alice: "I just flipped a coin [large number] times. Here's the sequence I got:
(Alice presents her sequence.)
Bob: No, you didn't. The probability of having gotten that particular sequence is 1/2^[large number]. Which is basically impossible. I don't believe you.
Alice: But I had to get some sequence or other. You'd make the same claim regardless of what sequence I showed you.
Bob: True. But am I really supposed to believe you that a 1/2^[large number] event happened, just because you tell me it did, or because you showed me a video of it happening, or even if I watched it happen with my own eyes? My observations are always fallible, and if you make an event improbable enough, why shouldn't I be skeptical even if I think I observed it?
Alice: Someone usually wins the lottery. Should the person who finds out that their ticket had the winning numbers believe the opposite, because winning is so improbable?
Bob: What's the difference between finding out you've won the lottery and finding out that your neighbor is a 500 year old vampire, or that your house is haunted by real ghosts? All of these events are extremely improbable given what we know of the world.
Alice: There's improbable, and then there's impossible. 500 year old vampires and ghosts don't exist.
Bob: As far as you know. And I bet more people claim to have seen ghosts than have won more than 100 million dollars in the lottery.
Alice: I still think there's something wrong with your reasoning here.