Scientific American has a brief article about the Sleeping Beauty Paradox. I think that thinking about it may be illuminating for the central importance of the idea of subjective probability:
So my analysis is this:
There’s a 50/50 chance that she’s in the one-time universe. And she knows she’ll be woken up, so waking her up conveys no information. It’s a bit like the show Severance, and so she's entitled to seeing her chances of being in each universe as 50-50.
But from a different perspective: imagine this: there are two staff members in the two-wakings universe, but neither of whom knows which universe they're in. Each one has an independent one-time assignment to wake her up. Because they don't know which universe they're in, they don't know if someone else will have or already has woken her up. (The day of the week thing doesn't matter: ignore it.) There's also one staff member in the one-waking universe who also doesn’t know which universe they’re in assigned to wake her up in. Each of the three -- on a bet -- should bet that they're in the two-wakings universe, since the odds are 2-1 that they are. Thus any particular moment of her waking up is more likely in the two-wakings universe than in the one-waking universe.
That seems obvious. On the other hand Sleeping Beauty gets no more info when she’s woken up than she had before, when it was 50-50 which universe she would be in. So there's no reason for her to believe that she's more likely to be in the two-wakings universe, since she'll be woken up either way, and each experience is completely independent of any others. There is no set of events that she can refer to. Her subjective experience (each time!) is to be woken up once and only once.
So if she sees herself from outside she’s going to bet that the person waking her up is one of the pair who don’t know which universe they’re in but who would rightly bet they’re in the two-wakings universe. But she has to take that circuit, relying on other people’s subjective probability, rather than her own. More simply: subjectively she has one experience of waking up, and there's a 50-50 chance that she's in the one-waking universe when she has that experience. But vicariously she knows that the person waking her up would rightly bet that they're in the two-wakings universe. And they should bet that they're in the two-wakings universe because each would know vicariously that anyone waking her up should bet on being in that universe. The point being that the wakers know that two other wakers also have the task of waking her up, whereas she's the only Sleeping Beauty, and will only have a memory of a single experience of being woken up. (There may be a tense logic to this: her memory is part of a present-tense subjective experience of her own past, whereas the wakers are having a present-tense objective experience of the objective existence in the present of other wakers.)
Anyhow, I think it's like the well-known two-envelopes problem:
There are two envelopes, one of which contains twice the amount of money as the other. You’re given a chance to switch. Is there an advantage to doing so?
The argument for it: It’s just as though you’re flipping a coin, where the stakes are putting half of what you have at risk to double the amount. I have $10; I flip a coin and have a 50% chance of getting $20, and a 50% chance or losing only $5. Of course that’s a good bet, Pascalians!
The argument against: I know one has $10 and one has $20. 50-50 chance I have the $20 envelope and 50-50 that I have the $10 envelope. If I have the $20 and I switch I’ll lose $10. If I have the $10 I’ll gain $10. 50-50 chance either way of gaining or losing $10.
The argument for switching hits a paradox because then why shouldn’t you switch again after you’ve already switched, and do this forever?
The argument against is clear cut and obviously true, even if you use $n and $2n without knowing what n is.
In the argument for switching there seem to be three equally possible amounts of money: n, 2n, n/2. And it’s presented as though all three are possible outcomes. But only two are possible, once the envelopes are sealed. Subjectively 3, but objectively 2. So this is where Bayesian subjective probability hits an infinite loop that frequentist probability wouldn’t.
The infinite loop can be thought of this way as well: every time j you're asked whether you want to switch the expected 1.5n that you got in the envelope last time now becomes a new nj. We're no longer talking about three possibilities being shoe-horned into 50% chance for each, but 4,5,6..., i.e: an infinite number of possibilities, each of which seems to have a 50% chance of being true at the moment that you consider it. So switching an infinite number of times should yield an infinite amount of money, but that's because you get stuck in a loop that can go on infinitely because the amount of money is indeterminate from the start.