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shminux comments on The Ultimate Sleeping Beauty Problem - Less Wrong Discussion
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Note that if she has no way to test whether her calculation is correct, the notion of probability does not make sense in her situation. In other words, what would she do differently if she estimates the probability to be, say, 1/2 instead of, say, 1/3?
If she was told she would be woken up 3^n times if n is even, 0 times if n is odd, then it seems obvious enough that when asked upon being woken up what she thought the probability that n is even, she would rationally and correctly say 100%. And that this would make sense. So Why wouldn't it make sense if the answer is some number other than 100%?
What she would do differently is bet on things she cared about based on the odds. Like "would you rather your relatives are given $5 if the number of coin flips is odd or $3 if the number of coinflips are even?" The answer for a rational beauty would depend on the probability that the number of coin flips is even.
What if after the experiment is over they'll tell her all the coin flip results, and have some bets resolved? Or they could do it each day after she answers. It wouldn't really change the interview, but it would change whether she can test her calculations.
I do not see how your first and second sentences are related. I might agree with the first sentence, but I do not agree with the second.
For example if I said "After this interview, before I put you to sleep, I will torture you with probability equal to the square of 1 minus the probability you give to the truth." If she estimates 1/2, she will say 1/2. If she estimates 1/3, she will say 1/3.
Edit: I interpreted your question as a rhetorical question implying that she would no do anything differently.
I should clarify. The universe doesn't ask you for numbers or ideas, it's your behavior that matters. Your example makes sense in a psychotic simulation, of course. Hopefully we are not living in one.
Ah, I think I understand your position now. You define "probability" as a measure of anticipation of outcomes (or possibly what her actions say about what outcomes she anticipate). If she is not anticipating learning whether or not the statement is true, then "probability" is not well defined.
Is this correct?
Well, one cannot always collect enough statistics to use a frequentist approach to check a Bayesian estimate. But one at least should be able to imagine possible worlds and, as you say, estimate measures of various outcomes, and make the best possible decision based on that. In the Sleeping Beauty problem there is no outcome and no decision to make.
So is the problem drastically different if after I ask you the interview question, I tell you how many times the coin was flipped? If so, assume that was the original problem.
How do we decide if your answer is correct? If you have all the power, you might as well just make up a random number between 0 and 1 and call it the answer. That's why the whole SSA vs SIA argument makes little sense.
Oops. You are right, what I said didn't make sense. I just edited the above post by changing "I tell you if you are right" to "I tell you how many times the coin was flipped"
OK, so, we can reformulate the question as "Dear Sleeping Beauty, what odds would you bet on the coin having been flipped an even number of times?", right? At least in this case the correctness of her answer can be explicitly tested by a frequentist simulation.
Sure, that is fine.
I am still curious though in the case where we do not reformulate the interview to that whether or not you think that the interviewer telling the beauty how many times the coin was flipped afterwords changes the question.