Sleeping Beauty problem

The Sleeping Beauty problem is a puzzle in probability theory: a sleeper is to be woken once or twice according to the toss of a coin, and asked her credence for the coin having come up heads.

The problem was originally stated by Adam Elga [ [http://www.princeton.edu/~adame/papers/sleeping/sleeping.html Self-locating belief and the Sleeping Beauty problem] by Adam Elga] but is based on earlier problems of imperfect recall [ [http://www.maproom.co.uk/sb.html "Sleeping Beauty" postings] ] and the older "paradox of the absentminded driver".

The problem

The paradox imagines that Sleeping Beauty volunteers to undergo the following experiment. On Sunday she is given a drug that sends her to sleep. A fair coin is then tossed just once in the course of the experiment to determine which experimental procedure is undertaken. If the coin comes up heads, Beauty is awakened and interviewed on Monday, and then the experiment ends. If the coin comes up tails, she is awakened and interviewed on Monday, given a second dose of the sleeping drug, and awakened and interviewed again on Tuesday. The experiment then ends on Tuesday, without flipping the coin again. The sleeping drug induces a mild amnesia, so that she cannot remember any previous awakenings during the course of the experiment (if any). During the experiment, she has no access to anything that would give a clue as to the day of the week. However, she knows all the details of the experiment.

Each interview consists of one question, "What is your credence now for the proposition that our coin landed heads?"

olutions

This problem is considered paradoxical because the answer is often given as either 1/3 or 1/2.

It seems obvious that the answer is 1/2. When Beauty wakes up she does not have any more or any less information than before the experiment. All she knows is that she has woken up and that this would have happened whether the coin landed heads or tails. She has no reason to believe that heads is more or less likely to have happened than tails. So the answer is 1/2. Or is it?

Let's examine it more carefully. Suppose this experiment were repeated 1,000 times. We would expect to get 500 heads and 500 tails. So Beauty would be awoken 500 times after heads on Monday, 500 times after tails on Monday, and 500 times after tails on Tuesday. In other words, only in a third of the cases would heads precede her awakening. So the right answer for her to give is 1/3.

This is the correct answer from Beauty's perspective. Yet to the experimenter the correct probability is 1/2. How do we reconcile the disparity between these two probability calculations? The reason there are two solutions is that they are solutions to two different questions. One is based on the percentage of runs of the experiment where the coin comes up heads, which is 1/2. The other is based on the percentage of interrogations where the coin comes up heads, which is 1/3. The way the original question is phrased determines the answer. In this case Sleeping Beauty is asked for her own credence, which is the percentage of interrogations where the coin comes up heads, so the answer is 1/3.

This still does not explain the disparity. The probability of an event may be defined as the theoretical knowledge that an observer has of the various paths leading to the present and their relative frequencies. The experimenter sees 1,000 paths leading to the present, consisting of 500 heads and 500 tails. Beauty sees not 1,000 but 1,500, of which 500 originate in heads and 1,000 in tails.

It may appear that the different probabilities, as determined by Beauty and the experimenter, are due to their different levels of knowledge. This is not so. Beauty's amnesia and ignorance of the day of the week are irrelevant. Even her waking up is irrelevant. The only factor that is relevant to the different probability calculations is sampling. If Beauty knew whether it was Monday or Tuesday then she would give the following odds. If it were Monday then the odds would be 1/2. If Tuesday then the odds would be 100% tails. Putting the two cases together we still get 1/3 chance of heads.

The deciding factor is not Beauty's lack of knowledge but how often each of the branches is sampled. By sampling the tails branch more times than the heads branch we guarantee that the probability of tails is higher than that of heads. To make it even clearer, suppose the setup was that Beauty would be awakened "only" after tails, not at all after heads. She would reply on being wakened that it had to be tails. This is the extreme case but it illustrates what is going on.

Another way to emphasize this effect is to increase the number of times, Sleeping Beauty is being awakened in the tails case. Imagine for instance, that she is still interviewed only once in the heads case, but interviewed e.g. a hundred times in the tails case, with a new case of amnesia after every interview. From her point of view, any interview will then be a hundred times more likely to be a part of a tails case than a heads case simply because more interviews take place during a tails case than during a heads case.

The core issue is whether Beauty is asked to estimate the probability of a certain outcome of the coin toss (which is assumed to be uniformly distributed) or the probability that a certain interview is a consequence of such a toss (which is skewed to the same extend as the number of samples).

This also applies to the experimenter, provided they are asked the question "at the same time" as Beauty. The reason why the experimenter's answer is given as 1/2 above is because it is tacitly assumed that the experimenter is being asked the question before the experiment or after it is over, but not during. This is the key difference that decides whether the probability of heads is 1/2 or 1/3. It is purely a matter of sampling. The answer of 1/3 arises simply because we sample twice as much on the tails branch.

Ambiguities

The question as posed by Elga was "When you are first awakened, to what degree ought you to believe that the outcome of the coin toss is Heads?" This is a question about the coin toss that precedes her awakening and is equivalent to asking "If you were to guess the result of the coin just tossed what chance is there that a guess of Heads would be correct?" which definitely has a 1/2 solution. However, the problem has been stated in various ways which has introduced ambiguity. In March 1999, before Elga's article, James Dreier had sent the problem to rec.puzzles with the question "What is your credence now for the proposition that our coin landed Heads?" With the enigmatic "credence", this could be interpreted as meaning "What probability should she give to the coin landing Heads in a run of the experiment?" and it is no longer clear if the question is one about her knowledge of the ratio of throws H:T that actually occur (or land) or one about her chance of guessing the throw that preceded her awakening. Of course, in variations of the problem, the crux of the matter lies in deciding which of the two questions is being asked.

Variations

The days of the week are irrelevant, but are included because they are used in some expositions. A non-fantastical variation called The Sailor's Child has been introduced by Radford Neal [ [http://arxiv.org/abs/math/0608592 Puzzles of Anthropic Reasoning Resolved Using Full Non-indexical Conditioning] Radford Neal] . The problem is sometimes discussed in cosmology as an analogue of questions about the number of observers in various different cosmological models.

The problem does not necessarily need to involve a fictional situation. For example computers can be programmed to act as sleeping beauty and not know when they are being run. For example consider a program that is run twice after tails is flipped and once after heads is flipped. If the program is set to always guess heads, it will be correct just 1/3 of the times if is run. If it is set to always guess tails, it will be correct 2/3 of the time it is run. So its credence should be to answer tails.

Only the knowledge of whether a coin was flipped or not has to be omitted. For example consider a man who flipped a coin yesterday. If it is heads he will flip the coin again today, then ask you if the last coin he flipped was heads or tails. If instead yesterday he flipped a tail, he won't flip today, instead he will just ask you if the last coin he flipped was heads or tails. You don't know if he flipped a coin today, but your credence should that half the time he flipped tails, and 1/4 of the time he reflipped heads and got tails, and 1/4 of the time he reflipped heads and got heads. So your credence is that there is 3/4 chance that he has last flipped tails.

References

ee also

*Credence — the subjective estimate of probability.

Other works discussing the Sleeping Beauty problem

* Arntzenius, F. (2002) Reflections on Sleeping Beauty, Analysis, 62-1, 53-62
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* Bradley, D. (2003) Sleeping Beauty: a note on Dorr's argument for 1/3, Analysis, 63, 266-268
* Dorr, C. (2002) Sleeping Beauty: in Defence of Elga, Analysis, 62, 292-296
* Elga, A. (2000) Self-locating Belief and the Sleeping Beauty Problem, Analysis, 60, 143-147
* Lewis, D. (2001) Sleeping Beauty: Reply to Elga, Analysis, 61, 171-176
* Meacham, C. (forthcoming) Sleeping Beauty and the Dynamics of De Se Beliefs, Philosophical Studies
* Monton, B. (2002) Sleeping Beauty and the Forgetful Bayesian, Analysis, 62, 47-53

External links

* [http://www.maproom.co.uk/sb.html Overview: Some "Sleeping Beauty" postings] — an archive containing many links, variants, formulations and arguments of thirders and halfers.
* [http://groups.google.com/groups?oi=djq&selm=an_455334937 Usenet posting by Jamie Dreier starting the discussion]
* [http://www.greylabyrinth.com/puzzles/puzzle.php?puzzle_id=puzzle075 "Sleeping Beauty" puzzle and solution]
* [http://dingo.sbs.arizona.edu/~thorgan/papers/Beauty.htm Terry Horgan: Sleeping Beauty Awakened: New Odds at the Dawn of the New Day] (review paper with references)
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*" [http://www.univ-corse.fr/~franceschi/Une%20application%20des%20n-univers.pdf Une application des n-univers à l'argument de l'Apocalypse et au paradoxe de Goodman] "