

Solomon Wright wrote: > Sorry, I missed most of the Sleeping Beaty thread, and most of the stuff I

> did catch seemed a bit complicated without the foundations. Can anyone

> point me in the right direction for decent, clear arguments for both the

> halfist and thirdist positions? (I've already seen the questions on Nick

> Wedd's page, what I want now is some answers to check my ideas against)

Here is something I posted a while back, explaining the (IMHO) three most convincing arguments for 1/3. Attached at the end are some details for the objective advisor setup. [From the postings of mine on Nick's page, you could probably tell that I was a halfer. I am now a thirder.] Conditionalization Argument

Summary: After waking, if Beauty learns that the day is Monday, then she should think P[heads]=1/2. (There is a simple argument for this based on the fact that knowledge of the day was exactly the piece of information she lost by the drug, and she thought P[heads] was 1/2 before she lost that information). On the other hand, if she learns that the day is Tuesday, then she should obviously think P[heads]=0. If she doesn't know the day, then the probability must lie somewhere in-between. Objective Advisor Argument

Summary: We can arrange for Beauty to meet with an objective advisor to discuss the probability of the coin toss. We can do it in such a way that the advisors presence can't influence Beauty's assessment of the coin toss, while the advisor, through Beauty being awake, does gain information supporting a particular outcome of the toss. Through standard probability techniques the advisor will compute that the probability of heads is only 1/3, and advise Beauty as such. If the situation is setup correctly, this can be the case even if Beauty and the advisor disclose every piece of information they have to each other. *For the record, all attempts to get Beauty to meet with an advisor who thinks that P[heads]=1/2 have resulted in Beauty getting new information. (and that information always allows her to agree with P[heads]=1/2) New Information Argument

Summary: The situation can be modified so that Beauty does seem to receive new information. Instead of waking her more times when the coin lands tails, instead wake her the same number of times regardless of the toss outcome, always without memory of any other wakening. Clearly, by symmetry, she should think that heads and tails are just as likely upon waking. So, suppose that we wait five minutes and then put her back to sleep unless the coin landed tails or it is her first day up. When she finds that she has not been put back to sleep she has new information which leads her to P[heads]=1/3. Further, the cases in which she is awake after five minutes are exactly the cases in the original problem. -- Details for the Objective Advisor argument -- One day, Sleeping Beauty visits the Institute of Memory Loss to be part of an experiment. She will be put to sleep on Sunday night, and a coin will be tossed. If the coin lands heads, then Beauty is woken either on Monday or Tuesday at random. On tails, she is woken on both Monday and Tuesday, subject to usual memory loss. In this experiment, Dr. Blue and Dr. Green, the world's foremost authorities on statistics, are hired to assist Beauty in her probability reasoning. If the coin landed heads, then one of the Statisticians is chosen randomly to help assist her. If the coin landed tails, then one of the PhDs gets her on one day, at random, and the other gets her on the other day. Now, using this setup, lets analyze the situation from Beauty's perspective in a random wakening. Clearly, after she wakes up and before she finds out which doctor she will visit, her probability assessments will be just as they are in the original Sleeping Beauty situation. Now, at that point, she knows that she will either meet with Dr. Blue or Dr.Green, but doesn't know which. Can her probability assessment change when she finds out which will advise her? Clearly not. Both men are, by symmetry, equivalent to her. One is not any more associated with heads or tails than the other. Hence, her probability assessment of the coin toss should be the same when she visits her advisor as it was when she woke. So, what will her advisor say about the probability of heads? Well, if the coin had landed heads, then he would only have had a 50% chance of meeting Beauty. On the other side of the coin, his meeting her would have been a certainty. Since he does get to meet her, he will believe that the probability of heads is only 1/3, when Beauty and him discuss the probability of heads in the coin toss, he will be quite insistent about this (you know how these Ph.D. types can be). As it stands, it is possible that the Statistician has been able to more accurately assess the situation with the piece of information that he has which she doesn't  he knows the day. Well, lets have him tell Beauty the day. How can this affect her probability assessment? Again, in this variant, it can't. By symmetry, neither Monday nor Tuesday is more indicative of heads or tails than the other. After this disclosure, Beauty and her advisor have exactly the same relevant information. It stands to reason that if they don't agree on the probabilities at this point, then one of them is wrong.