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I can no longer look at Nick Bostrom the same way after finding out he’s a halfer in the Sleeping Beauty Problem.
We actually had a discussion about this about 5 months ago, and I still stand by my response from then. The only thing that's possibly to his discredit is thinking that "the Sleeping Beauty Problem" has a well-defined answer at all. If you are not willing to commit to something like "at each awakening, Beauty makes a bet at these odds, what will maximise her earnings?", your question for "the probability" might as well be asking about a dog's Buddha nature.
I guess if you want to get metaphysical about the idea of probability and statistics at all then you can dismiss the problem as underspecified, but you'd have to apply the same radical skepticism to coin flip odds, dice rolls, and all other stochastic phenomena. I mean, can you really say that a coin has a 50% chance of coming up heads? Any given coin flip is a completely determanistic process, only capable of coming up heads, or only capable of coming up tails. Probability is just a shorthand we use to make sense of pseudorandomness beyond our control or comprehension.
When one uses the same tools of probability and statistics that we use to analyze every other stochastic phenomenon to analyze the Sleeping Beauty problem, one finds that there are three equally likely states that are indestinguishable from Beauty's point of view. Two of them correspond to tails, and the other one corresponds to heads. The math is simple.
The underspecification is in the conversion of the word problem to a rigorously defined sample space, not in the interpretation of what happens once you have done that. I am not aware of any "tools of probability and statistics" worth the name that would help with this. A "1/2er" presumably would insist that the question Beauty is asked (like "what is the probability that the coin landed Heads?") is about a sample space with two states (coin landed H or T). If you want, you can think of it as a sort of repeatability
Here are two specific ways to refine the problem (replacing the ambiguous question for Beauty's "probability" with well-defined bets, vaguely inspired by Dutch book arguments): say Beauty is asked for odds that she will accept for a bet on the coin's state. A bet on H at odds 1:n will be executed as follows: Beauty pays $1, and gets back $(1+n) iff the coin is H.
Version 1: Beauty is asked every time she is woken up, and the bet is played immediately upon asking her. What is the lowest n such that she can bet on H at odds 1:n and not have a loss in expectation? (The answer is 2, corresponding to 1/3 "probability".)
Version 2: Beauty is asked every time she is woken up. At the end of the experiment (so after 1 or 2 awakenings), we take her answer unless we asked twice and she contradicted herself (which she anyhow can only do if she randomised), and then play the bet once. What is the lowest n such that she can bet on H at odds 1:n and not have a loss in expectation? (The answer is 1, corresponding to 1/2 "probability".)
You can protest that refining the problem statement into Version 2 rather than Version 1 defies common sense, but I don't think you can argue that it defies "the tools of probability of statistics that we use to analyze every other stochastic phenomenon".
You cannot ask her this question. You literally cannot ask this of her, because any question you ask of a person is automatically attached to the modifier "conditional on the fact that I am asking you this question", which here splits it into three cases. The only way for Beauty to not rationally update on the fact that you asked a question is if you either don't entangle your asking on any of the results of the coin flip, or if you lie to her about the premises of the problem, in which case she can be dutch booked and believe in incorrect probabilities like 1/2 because she's been deceived.
It absolutely is defying those tools, because you are combining multiple answers into a single bet. You're essentially weighting bets based on the outcome. Consider
Version 3: Beauty never goes to sleep or is woken up or has any amnesia, she's just a normal person. A bookie flips a coin weighted to come up heads 1/3 of the time (according to normal probability rules) and then flips a second coin, this one fair 50-50. If both coins are head He tells her about the first coin and asks her to bet if it's heads or tails at 1:n odds. If the first coin actually is heads, the bookie pays out normally. If the first coin was tails he looks at the second coin, and if it's also tails he pays out the bet, but if it's heads he reneges on the deal and runs away, not taking her money nor paying her (though she would have lost betting on heads). Beauty can now bet on heads at 1:1 odds with no loss, but this does not correspond to a 50-50 probability that the coin actually lands heads because the declared payouts are not honest. Half the time she bets heads and would lose she doesn't lose anything, so she can bet heads more freely. She's betting on "the ratio of the probability you will take my money to the probability you will not take my money". What you want is "the probability the coin came up heads, conditional on you asking me this question right now and me making this bet". For normal betting procedures we make sure these are equal and can thus use them interchangeably, but your version 2 disentangles them.
Mathematically, this is equivalent to your version 2. This is why you get the answer for your "bets" and the actual probability diverging, because half of her bets are being cancelled/fused. In the tails scenario you asked her twice, she bet and lost twice, but you only took her money once.
This does not track with real-life language usage; otherwise, a whole swath of common expressions ranging from the colloquial to at least semi-formal settings would be rendered incoherent. "I got a middle seat on my Spirit Airways flight, and it turned out I was sandwiched between Avril Lavigne and Justin Bieber, and they didn't know about each other's plans! What's the likelihood of this happening?" (Fairly high, conditional on the fact that someone would come up with that scenario and ask the question seriously?)
For examples that are more in the class of mathematical word problems, something like shuffling a deck of cards, getting an unexpected outcome (e.g. the first 10 cards are all the same suit) and then asking about the probability of it is a very common idiom, and always understood to refer to the probability of that outcome if the experiment were repeated. This is actually very similar to the "1/2er" interpretation of the Sleeping Beauty problem I suggested. Only contrarian weirdos like me even get the idea of taking the anthropic-principle angle towards that question, and rebutting with the question what subset of permutations the original asker would consider sufficiently "special" to be surprised by and ask; but this is what I would need to do, if I seriously wanted to answer the question "what is the probability of the first 10 cards being all the same suit, conditional on me asking this question"! (Most shuffles would presumably be boring and you would not ask anything about them!)
I'm unfortunately struggling to parse your "Version 3" example. You say
Is the "If both coins are head" a leftover from some previous edit? Because otherwise, it seems that the "If the first coin was tails" condition can never be met (he only asks her to bet if both coins came up heads?). Also, even removing that phrase, I don't understand the meaning of "He tells her about the first coin and asks her to bet (...)". What does it mean that "he tells her about the first coin"? Is it just that he tells her "I just flipped a biased coin with 1/3 P(heads)"?
(...but either way, even if the example you are intending to communicate is mathematically equivalent to my Version 2, the existence of a contrived mathematically equivalent construction is not proof that the original construction is contrived! You can build contrived isomorphic setups for any setup.)
On the first point, you're right that it is possible to ask this question. I suppose I exaggerated what I was trying to say. The issue I think is language tense. If you ask in the progressive tense "what are the odds of this happening, then you are asking someone about repeated probabilities. "If I, knowing nothing, get on a plane, what are the odds of A and B happen simultaneously?" The correct answer would be to compute the probability of A, the probability of B, and then multiply them together. Because you're not asking about whether this happened in the real world, but about whether it could/would happen in general.
If you ask in the past tense "what are the odds that this happened, this is a question about the world. This is actually the question "What are the odds that this thing happened, conditional on everything you know right now, including me asking you this question?" It is not a question about general repeated probabilities, because that's not how verb tenses work. It's past tense. You could convert it into a question about repeated probabilities (which you might need to if you are a frequentist), but if you did it would translate into "What are the odds of this thing happening conditional on you finding yourself in a mathematically analogous situation to the one you find yourself in now." If you ask me the probability that you yourself were sandwiched between Avril Lavigne and Justin Bieber on a flight I'm not going to compute the probability of them being on flights, I'm going to say ~0% because if that had actually happened you would have phrased it very differently when using it as an example.
You're also right that I mangled my example while editing. The example is supposed to create a scenario where there's a 50% everything is normal (we flip one coin and it's heads) a 50% chance we have a flaky bookie (who in turn has a 50% chance of reneging on his bet). The point is not that the example is "contrived", the point is that it detaches betting odds from probabilities because the payouts are distorted. Consider a friend who, on a first roll, fumbles his dice and drops them clumsily. If the result is a 1 he says it doesn't count and rerolls them properly, keeping the result no matter what. But if the fumbled roll is good he keeps it. If this were a consistent pattern you would be on his dice differently than 1/6 per side, because you're not betting on the probability that a die rolls a certain number in a vacuum, but the probability that a certain number is kept in the end.
When sleeping Beauty wakes and makes a bet, there's a chance your version 2 is going to discard her bet and roll again, only accepting her bet if she wakes up and makes the same bet again the next day. If she always bets on "heads" she will be wrong 2/3 of the time she says it, but lose money 1 time and gain money 1 time. You might as well never wake her up on Tuesday at all because you're essentially taking bets on Monday in both cases and then ignoring her Tuesday answer unless it conflict with Monday. The probability you're actually getting here is "Conditional on me asking you this question and this being a day when your answer actually matters for betting purposes, what is the probability of it being heads?" which is a very very different question from "what is your belief that the coin is heads right now?" which is what she's actually asked in the original question.
I think a more straightforward way to notice that this scenario detaches P(heads|you just woke up) from the optimal betting strategy is to compare it to the following scenario:
Some researchers flip a coin without showing you the result. On Monday, they interview you about the coin and ask you to make bets about its status. Then, on Tuesday, if the result was tails, the researchers play the videotape of your interview from Monday and perform all your bets a second time on your behalf.
Here, your belief that the coin landed on tails should clearly be 0.5 even given the condition that you're currently being interviewed. But if you make any bets, you need to keep in mind that they'll be executed twice in the tails condition. The optimal strategy is the same as in the original Sleeping Beauty problem, since that problem supposes that you were going to do the same thing on both days anyway. (That strategy is not as straightforward as "assign probability 0.6667 to tails" if you can bet things like "all the money currently in my checking account" rather than just fixed dollar amounts.)
So within the problem, the concept of "credence" is not as broadly applicable as it normally is; the conditional probability is different from the optimal betting odds (and those odds themselves differ based on details of the bet). You can either stick to the conditional-probability definition, say that the odds are 0.5 (0.6667 in the original problem), and not use that value for any practical purpose. Or you can say "I think there is a 50% chance that the coin is tails, and if that is the case any actions I take will happen twice", which is a more useful fact to know when strategizing.
I think you detached them in the opposite way here. In the original problem both the conditional probability and optimal betting odds are 0.6667. In /u/4bpp version (and the version I attempted to describe) the conditional probability is still 0.6667 but the optimal betting odds go to 0.5. In your version the conditional probability is 0.5 and the optimal betting odds are 0.6667. You are correct that this is an easier way to describe how betting odds and conditional probabilities can detach.
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