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shminux comments on The Ultimate Sleeping Beauty Problem - Less Wrong Discussion
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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.
Well, there are two issues there, one is the divergent weights given to the lower-probability flip sequences (the St. Petersburg paradox), the other is the meaning of the term "subjective probability". Asking for the odds gives a concrete interpretation to the latter. As for the former, you can probably get any answer you want, depending on how you choose to sum the divergent series.