The Sleeping Beauty problem is a puzzle in probability theory and formal epistemology in which an ideally rational epistemic agent is to be wakened once or twice according to the toss of a coin, and asked her degree of belief for the coin having come up heads.
The problem was originally formulated in unpublished work by Arnold Zuboff (this work was later published as "One Self: The Logic of Experience"[1]), followed by a paper by Adam Elga[2] but is based on earlier problems of imperfect recall and the older "paradox of the absentminded driver".
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Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details. On Sunday she is put to sleep. A fair coin is then tossed 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 and Tuesday. But when she is put to sleep again on Monday, she is given a dose of an amnesia-inducing drug that ensures she cannot remember her previous awakening. In this case, the experiment ends after she is interviewed on Tuesday.
Any time Sleeping beauty is awakened and interviewed, she is asked, "What is your credence now for the proposition that the coin landed heads?"
This problem continues to produce ongoing debate.
The thirder position argues that the probability of Heads is 1/3. Adam Elga argued for this position originally.[2] His argument is as follows. Suppose Sleeping Beauty is told and she comes to fully believe that the coin landed tails. By a restricted principle of indifference, her credence that it is Monday should equal her credence that it is Tuesday since being in one situation would be subjectively indistinguishable from the other.
Consider now that Sleeping Beauty is told upon awakening and comes to fully believe that it is Monday. She is guided by the objective chance of heads landing being equal to the chance of tails landing. Thus,
P(Tails and Tuesday) = P(Tails and Monday) = P(Heads and Monday)
Since these three outcomes are exhaustive and exclusive for one trial, the probability of each is one third by the previous two steps in the argument.
Another argument is based on long run average outcomes. 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. This long run expectation should give us the same expectations for the one trial, so P(Heads)=1/3.
Nick Bostrom argues that the Thirder position is implied by the Self-Indication Assumption.
David Lewis responded to Elga's paper with the position that Sleeping Beauty's credence that the coin landed heads should be 1/2.[3] Sleeping Beauty receives no new non-self-locating information throughout the experiment because she is told the details of the experiment. Since her credence before the experiment is P(Heads)=1/2, she ought to continue to have a credence of P(Heads)=1/2 since she gains no new relevant evidence when she wakes up during the experiment.
Nick Bostrom argues that the The Halfer position is implied by the Self-Sampling Assumption.
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. The problem is sometimes discussed in cosmology as an analogue of questions about the number of observers in various 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.
This differs from the original in that there are one million and one wakings if tails comes up. It was formulated by Nick Bostrom.[4]