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Notes -
(I mean mathematically correct in the sense that Kolmogorov isn't technically the only game in town with internal axiomatic consistency, though it's universal enough in use I was probably being overly pedantic there)
Because Monty Hall is inherently grounded, while Sleeping Beauty is a weird contrivance pretty much on purpose. Sleeping Beauty relies on a supposed perfect memory-erasing amnesia drug erasing one entire interview and only that one interview. It further relies on Beauty being unable to distinguish the passage of time at all, and even more confusingly we are including Beauty's answers across multiple days in our sample space! This is unintuitive. Our sample space to get 1/3 is: Beauty on Monday on Heads, Beauty on Monday on Tails, Beauty on Tuesday on Tails, yes? Most probability problems are not so casual about employing asymmetric tree diagrams across temporal positions, because the eminently natural assumption about the passage of time is that you were able to perceive it. The weird, nonexistent mind-altering drug breaks that intuition about the unbroken forward flow of time! An assumption we virtually never question in any other scenario.
So despite my best wishes I guess I'll take the bait. To be clear, I'm not so much trying to explain the halfer position as elucidating why I believe the whole debate to be kind of stupid and misguided, though I am quite sympathetic to your view.
Anyways, time flow. In other words, the halfer position rejects that it even makes sense to ask about Beauty on Tuesday, since "obviously" the sample space is only: Beauty on Monday with two possible coin flip results (i.e. guesses). The halfer position says in effect that it's impossible to consider two super-imposed Tails-guessing Beauties on both Monday and Tuesday at once. Or, phrased a different (and probably better) way, a Monday Beauty guessing tails is functionally indistinguishable from a Tuesday Beauty guessing tails, because the "divergence" in intent has already occurred! The only relevant guess is the coin.
The second illuminating follow-up question: What is our reward scheme? Do we reward Beauty for a correct answer every time she wakes up (and then steal it back when she sleeps and forgets, thus making any gain ephemeral; though optionally we may choose to sum all three of her choices for aggregate statistical reasons), or do we reward Beauty only after it's Wednesday? For the former, we are effectively rewarding each awakening, but for the latter we provoke a philosphical crisis. Is Tuesday Beauty really making a truly independent choice? Halfers might say no, of course not, "reality" already diverged. Thirders would say yes, of course, it's a new day so thus a new choice. Crisis aside, consider a Beauty who goes "screw it, I'm not playing mind games, I'm choosing heads literally every time" - for a one-time Wednesday-only reward, she wins half the time. Can we truly treat a Beauty who goes "screw it, I'm choosing Tails every time" differently? It depends on our reward scheme! In one setup it's clear this Tails-stubborn Beauty gets double winnings every Wednesday (because even though both awakenings gave the same answer, they were rewarded separately thus double dipping), while in the other she is no better off than the Heads-stubborn one (because the coin was, in fact, tails just half the time, and she's only rewarded at the end). Hopefully that teases apart why it matters.
But you see the issue here, previously obscured? Not only is this contrived, but we require some clarification here about definitions to deliver an answer. We could use a computer, but then we're merely revisiting the same problem with our programming as a design choice: when the coin comes up Tails, do Monday-Beauty and Tuesday-Beauty execute their decision-making code twice with independent randomness, or does Tuesday-Beauty simply output the duplicated cached result from Monday? We implicitly make a claim, one of the following:
This whole setup is odd, because typically in a probability problem, identical epistemic states with identical available information should have identical probability outputs/beliefs, right? Yet in one of these cases, we're saying the two events are separate because 'someone said so'. Or maybe more accurately, in one case we're talking about epistemic states of knowledge, and in the other we're talking about specific events. Scope is subtly different. The problem has laundered in a sneaking modeling choice without you realizing it. Your choice of model literally determines if additional randomness is injected into the system or not, and thus influences the long-run probability you will find. This is especially clear when you add simple rewards like I described.
But anyways real life does not contain weird situations like these reminiscent of quantum physics. Monty Hall can be modeled strictly mechanically, and in a loose sense so can Sleeping Beauty... but how you represent said model is not a settled question. Is the experiment truly "reset" when we move from Monday to Tuesday? Again that's really a purely philosophical question, not a mathematical one. The presence of a belief-having chooser like Beauty is required for us to even talk about "beliefs" and "rational bets" and all that stuff. This is the doubly case when it comes to time. It's one of the most frustrating aspects of statistics and probability: we cannot actually run perfectly authentic, true counterfactuals, because time runs in one direction. Just like science fiction can only theorize and imagine what would happen in multiverses or if we perfectly cloned a human mind, probability also struggles to perfectly map to reality and human perception because of the aforementioned triple concept divergence in what we mean when we say "probability".
Maybe I'm being too harsh on this thought experiment, but I have little patience for them when they so obviously diverge from reality. We shouldn't be surprised that setting up an unintuitive situation produces unintuitive answers.
I think I'm Sleeping Beauty'd out, but thanks for your comments. I honestly don't think the problem's all that existentially weird - compared to many thought experiments, this one could at least take place in our physical universe.
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