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 hypotheses or world-models M, and denote a member of this space as M'. Given data D, you can update using Bayes' Theorem and obtain a posterior distribution. We can quantify the difference between the prior and posterior using the Kullback-Leibler divergence and use it as a measure of Bayesian surprise. To see how one thing with the same information content as another thing can be more or less surprising, imagine that we have an agent using this framework set in front of a television screen broadcasting white noise. The information content of each frame is very high because there are so many equally likely patterns of noise, but the agent will quickly stop being surprised because it will settle on a world-model that predicts random noise, and the difference between its priors and posteriors over world-models will become very small.
If in the future we want to keep using a coin flip example, I suggest forgetting things that are so mind-like as 'Alice is clairvoyant', and maybe just talk about biased and unbiased coins. It seems like an unnecessary complication.
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?
Because the number of bits of information is the same in all cases. Any given random sequence provides evidence of countless extremely low probability world models - we just don't consider the vast majority of those world-models because they aren't elevated to our attention.
...If in the future we want to keep using a coin flip example
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.