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Michaelos comments on Fundamentals of kicking anthropic butt - Less Wrong
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Manfred, thanks for this post, and for the clarifications below.
I wonder how your approach works if the coin is potentially biased, but the bias is unknown? Let's say it has probability p of Tails, using the relative frequency sense that p is the frequency of Tails if tossed multiple times. (This also means that in multiple repetitions a fraction 2p / (1 + p) Beauty awakenings are after Tails, and a fraction 1 / (1 + p) Beauty awakenings are on Mondays.)
Beauty has to estimate the parameter p before betting, which means in Bayesian terms she has to construct a subjective distribution over possible values of p.
Before going to sleep, what should her distribution look like? One application of the indifference principle is that she has no idea about p except that it is somewhere between 0 and 1, so her subjective distribution of p should be uniform on [0, 1].
When she wakes up, should she adjust her distribution of p at all, or is it still the same as at step 1?
Suppose she's told that it is Monday before betting. Should she update her distribution towards lower values of p, because these would give her higher likelihood of finding out it's Monday?
If the answer to 3 is "yes" then won't that have implications for the Doomsday Argument as well? (Consider the trillion Beauty limit, where there will be a trillion awakenings if the coin fell Tails. In that case, the fraction of awakenings which are "first" awakenings - on the Monday right after the coin-toss - is about 1/(1 + 10^12 x p). Now suppose that Beauty has just discovered she's in the first awakening... doesn't that force a big shift in her distribution towards p close to zero?)
I may be confused, but it seems like Beauty would have to ask "Under what conditions am I told 'It's Monday'?" to answer question 3.
In other problems, when someone is offering you information, followed by a chance to make a decision, if you have access to the conditions under which they decided to offer you that information should be used as information to influence your decision. As an example, the other host behaviors in the Monty Hall problem. mention that point, and it seems likely they would in this case as well.
If you have absolutely no idea under what circumstances they decided to offer that information, then I have no idea how you would aggregate meaning out of the information, because there appear to be a very large number of alternate theories. For instance:
1: If Beauty is connected to a random text to speech generator, which happens to randomly text to speech output "Smundy", Beauty may have misheard nonsensical gibberish as "It's Monday."
2: Or perhaps it was intentional and trying to be helpful, but actually said "Es Martes" because it assumed you were a Spanish speaking rationalist, and Beauty just heard it as "It's Monday." when Beauty should have processed "It's Tuesday." which would cause Beauty to update the wrong way.
3: Or perhaps it always tells Beauty the day of the week, but only on the first Monday.
4: Or perhaps it always tells Beauty the day of the week, but only if Beauty flips tails.
5: Or perhaps it always tells Beauty the day of the week, but only if Beauty flips heads.
6: Or perhaps it always tells Beauty the day of the week on every day of the puzzle, but doesn't tell Beauty whether it is the "first" Monday on Monday.
7: It didn't tell Beauty anything directly. Beauty happened to see a calendar when it opened the door and it appears to have been entirely unintentional.
Not all of these would cause Beauty to adjust the distribution of P in the same way. And they aren't exhaustive, since there are far more then these 7. Some may be more likely than others, but if Beauty don't have any understanding about which would be happening when, Beauty wouldn't know which way to update P, and if Beauty did have an understanding, Beauty would presumably have to use that understanding.
I'm not sure whether this insightful, or making it more confused then it needs to be.
OK, fair enough - I didn't specify how she acquired that knowledge, and I wasn't assuming a clever method. I was just considering a variant of the story (often discussed in the literature) where Beauty is always truthfully told the day of the week after choosing her betting odds, to see if she then adjusts her betting odds. (And to be explicit, in the trillion Beauty story, she's always told truthfully whether she's the first awakening or not, again to see if she changes her odds). Is that clearer?
Yes, I wasn't aware "Truthfully tell on all days" was a standard assumption for receiving that information, thank you for the clarification.
It's OK.
The usual way this applies is in the standard problem where the coin is known to be unbiased. Typically, a person arguing for the 2/3 case says that Beauty should shift to 1/2 on learning it is Monday. Whereas a critic originally arguing for the 1/2 case says that Beauty should shift to 1/3 for Tails (2/3 for Heads) on learning it is Monday.
The difficulty is that both those answers give something very presumptuous in the trillion Beauty limit (near certainty of Tails before the shift, or near certainty of Heads after the shift).
Nick Bostrom has argued for a "hybrid" solution which avoids the shift, but on the face of things looks inconsistent with Bayesian updating. But the idea is that Beauty might be in a different "reference class" before and after learning the day.
See http://www.fhi.ox.ac.uk/__data/assets/pdf_file/0011/5132/sleeping_beauty.pdf or http://www.nickbostrom.com/ (Right hand column, about halfway down the page).