Sleeping Beauty problem
Encyclopedia
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The problem was originally formulated in unpublished work by Arnold Zuboff (this work was later published as "One Self: The Logic of Experience"), followed by a paper by Adam Elga but is based on earlier problems of imperfect recall and the older "paradox of the absentminded driver".
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?"
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
.
responded to Elga's paper with the position that Sleeping Beauty's credence that the coin landed heads should be 1/2. 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 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.
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
and formal epistemology
Formal epistemology
Formal epistemology uses formal methods from decision theory, logic, probability theory and computability theory to elucidate epistemic problems. Work in this area spans several academic fields, including philosophy, computer science, economics, and statistics...
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"), followed by a paper by Adam Elga but is based on earlier problems of imperfect recall and the older "paradox of the absentminded driver".
The problem
Sleeping BeautySleeping Beauty
Sleeping Beauty by Charles Perrault or Little Briar Rose by the Brothers Grimm is a classic fairytale involving a beautiful princess, enchantment, and a handsome prince...
volunteers to undergo the following experiment and is told all of the following details. On Sunday she is put to sleep. A fair coin
Fair coin
In probability theory and statistics, a sequence of independent Bernoulli trials with probability 1/2 of success on each trial is metaphorically called a fair coin. One for which the probability is not 1/2 is called a biased or unfair coin...
is then tossed
Coin flipping
Coin flipping or coin tossing or heads or tails is the practice of throwing a coin in the air to choose between two alternatives, sometimes to resolve a dispute between two parties...
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?"
Thirder position
The thirder position argues that the probability of Heads is 1/3. Adam Elga argued for this position originally. 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
Self-Indication Assumption
The Self Indication Assumption Nick Bostrom originally used the term SIA in a slightly different way. What is here referred to as SIA, he referred to as the combined SSA+SIA, a philosophical principle defined by Nick Bostrom, one of the two major schools of anthropic probability , states that:Note...
.
Halfer position
David LewisDavid Kellogg Lewis
David Kellogg Lewis was an American philosopher. Lewis taught briefly at UCLA and then at Princeton from 1970 until his death. He is also closely associated with Australia, whose philosophical community he visited almost annually for more than thirty years...
responded to Elga's paper with the position that Sleeping Beauty's credence that the coin landed heads should be 1/2. 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
Self-Sampling Assumption
The self-sampling assumption , one of the two major schools of anthropic probability , states that:...
.
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. 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.
Extreme Sleeping Beauty
This differs from the original in that there are one million and one wakings if tails comes up. It was formulated by Nick Bostrom.Other works discussing the Sleeping Beauty problem
- Arntzenius, F. (2002) Reflections on Sleeping Beauty, Analysis, 62-1, 53-62
- 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
- R. Neil, Puzzles of Anthropic Reasoning Resolved Using Full Non-indexical Conditioning, preprint
- Zuboff, M. (1990) One Self: The Logic of Experience', Inquiry, 33, 39-68
External links
- Terry Horgan: Sleeping Beauty Awakened: New Odds at the Dawn of the New Day (review paper with references)
- Anthropic Preprint Archive: The Sleeping Beauty Problem: An archive of papers on this problem
- Phil Papers Entry on Sleeping Beauty (a complete bibliography of papers on the problem)