Man, that sleeping beauty digression at the end. I'm so easily aggroed.
I wonder if I should write up a rant or a better attempt at exposition, against this "clearly the problem is underdetermined" position. The correct answer is somewhat difficult and all expositions I've seen (or written!) so far have had big shortcomings. But even if one didn't know the right answer, one should readily conclude that a principled answer exists. We assign things probabilities for fairly deep reasons, reasons undisturbed by trifles like the existence of an amnesia pill.
That aside, good talk :)
Hi Charlie! Actually I complete agree with Vladimir on this: subjective probabilities are meaningless, meaningful questions are decision theoretic. When the sleeping beauty is asked "what day is it?" the question is meaningless because she is simultaneously in several different days (since identical copies of her are in different days).
Vladimir Slepnev (aka cousin_it) gives a popular introduction to logical counterfactuals and modal updateless decision theory in the Tel Aviv LessWrong meetup.
[https://www.youtube.com/watch?v=Ad30JlVh4dM&feature=youtu.be]