I’m having a really hard time pinpointing where there’s an error in the analysis, but something is still just not right. There is no indexical uncertainty in locating the event of the coin being flipped on Tuesday. Any information relevant to that event can only be considered information if there is a different probability of it being received if the coin is heads rather than tails. Any stream of bits that Beauty receives has the same probability no matter what that event is. So her probability of that event simply cannot be updated in any direction. Where does this reasoning go wrong?
On the contrary, the probability that Beauty has some particular set of Monday/Tuesday experiences is twice as great if she is woken on both days than if she is woken only on Monday (assuming that the probability in any case is very small, as it will be for any ordinary set of waking experiences).
Have you looked at my Sailor’s Child problem (in the referenced paper)? It is intended to be completely analogous to the only-mildly-fantastical (ie, original) version of Sleeping Beauty, while being totally non-fantastical. I assume that agreement can be obtained on completely non-fantastical problems, and I think the answer for Sailor’s Child is clearly 1⁄3, not 1⁄2, so if it is indeed analogous to Sleeping Beauty, that shows that 1⁄3 is the correct answer there as well. If you think it is not analogous, then in what relevant way is it different?
My paper also lists various other variations on Sleeping Beauty (eg, introducing a “Prince”), which also seem to me to definitively establish that the correct answer is 1⁄3. Plus there are the betting arguments, including one I talk about in my reply to Part 1 of this post. They also see definitive to me, unless you are willing to turn “probability” into something that isn’t a guide to decision-making.
On the contrary, the probability that Beauty has some particular set of Monday/Tuesday experiences is twice as great if she is woken on both days than if she is woken only on Monday
Ok, this sentence made everything snap into place for me. Thanks. The Sailor’s Child problem is also helpful. This has been an interesting journey. I was originally a one-thirder based on betting arguments, and then became convinced from the original post that that is indeed a red herring, and so momentarily became a halfer, and now that you’ve clarified this I’m back to being a thirder to within epsilon.
I’m having a really hard time pinpointing where there’s an error in the analysis, but something is still just not right. There is no indexical uncertainty in locating the event of the coin being flipped on Tuesday. Any information relevant to that event can only be considered information if there is a different probability of it being received if the coin is heads rather than tails. Any stream of bits that Beauty receives has the same probability no matter what that event is. So her probability of that event simply cannot be updated in any direction. Where does this reasoning go wrong?
On the contrary, the probability that Beauty has some particular set of Monday/Tuesday experiences is twice as great if she is woken on both days than if she is woken only on Monday (assuming that the probability in any case is very small, as it will be for any ordinary set of waking experiences).
Have you looked at my Sailor’s Child problem (in the referenced paper)? It is intended to be completely analogous to the only-mildly-fantastical (ie, original) version of Sleeping Beauty, while being totally non-fantastical. I assume that agreement can be obtained on completely non-fantastical problems, and I think the answer for Sailor’s Child is clearly 1⁄3, not 1⁄2, so if it is indeed analogous to Sleeping Beauty, that shows that 1⁄3 is the correct answer there as well. If you think it is not analogous, then in what relevant way is it different?
My paper also lists various other variations on Sleeping Beauty (eg, introducing a “Prince”), which also seem to me to definitively establish that the correct answer is 1⁄3. Plus there are the betting arguments, including one I talk about in my reply to Part 1 of this post. They also see definitive to me, unless you are willing to turn “probability” into something that isn’t a guide to decision-making.
Ok, this sentence made everything snap into place for me. Thanks. The Sailor’s Child problem is also helpful. This has been an interesting journey. I was originally a one-thirder based on betting arguments, and then became convinced from the original post that that is indeed a red herring, and so momentarily became a halfer, and now that you’ve clarified this I’m back to being a thirder to within epsilon.