Sleeping Beauty problem

The problem was originally formulated in unpublished work in the mid 1980s by Arnold Zuboff (the work was later published as "One Self: The Logic of Experience")[1] followed by a paper by Adam Elga.[2] A formal analysis of the problem of belief formation in decision problems with imperfect recall was provided first by Michele Piccione and Ariel Rubinstein in their paper: "On the Interpretation of Decision Problems with Imperfect Recall" where the "paradox of the absent minded driver" was first introduced and the Sleeping Beauty problem discussed as Example 5.[3][4] The name "Sleeping Beauty" was given to the problem by Robert Stalnaker and was first used in extensive discussion in the Usenet newsgroup rec.puzzles in 1999.[5]

Sleeping Beauty volunteers to undergo the following experiment and is told all of the following details: On Sunday she will be put to sleep. Once or twice, during the experiment, Sleeping Beauty will be awakened, interviewed, and put back to sleep with an amnesia-inducing drug that makes her forget that awakening. A fair coin will be tossed to determine which experimental procedure to undertake:

• If the coin comes up heads, Sleeping Beauty will be awakened and interviewed on Monday only.
• If the coin comes up tails, she will be awakened and interviewed on Monday and Tuesday.

In either case, she will be awakened on Wednesday without interview and the experiment ends.

Any time Sleeping Beauty is awakened and interviewed she will not be able to tell which day it is or whether she has been awakened before. During the interview Sleeping Beauty is asked: "What is your credence now for the proposition that the coin landed heads?"

This problem continues to produce ongoing debate.

To resolve the paradox it is unavoidable that we must first define and interpret what probability means. The frequentist interpretation is that probability measures the emergent pattern of many samples; the Bayesian interpretation is that probability measures the subjective belief of the experimenter or in this case the subject.

Let ${\displaystyle X}$ be the random variable governing the fair coin toss, where ${\displaystyle X=1}$ (heads) with a long-run limiting proportion ${\displaystyle \theta =1/2}$. If we adopt a frequentist interpretation, then the long-run limiting proportion of heads for the fair coin in this investigation, ${\displaystyle \theta =1/2}$, is the probability of heads regardless of who is perceiving the flip and when. If we adopt a Bayesian interpretation then "probability of heads" initially references the prior predictive distribution for the coin landing heads,

${\displaystyle P(X=1)=\int P(X=1|\theta =c)\cdot dP(\theta \leq c)}$,

based on perfect knowledge that the limiting proportion of heads is ${\displaystyle \theta =1/2}$ with belief probability ${\displaystyle P(\theta =1/2)=1}$. To the Bayesian the limiting proportion of heads, ${\displaystyle \theta }$, is not a probability. Sleeping Beauty's belief, ${\displaystyle P(X=1)}$, is the probability which can later change. If Sleeping Beauty had imperfect knowledge of ${\displaystyle \theta }$ then ${\displaystyle P(\theta \leq c)}$ would not be a degenerate distribution and ${\displaystyle P(X=1)}$ would represent her belief in ${\displaystyle X=1}$ with imperfect knowledge of ${\displaystyle \theta }$.

Using Bayes' theorem, Sleeping Beauty's prior predictive distribution for heads is updated based on the information available to her about the number of possible awakenings (1 or 2), forming a posterior predictive distribution. This results in a predictive belief probability of ${\displaystyle 1/3}$ that the coin landed heads,

${\displaystyle P(X=1|{\text{awakening}}=1\cup {\text{awakening}}=2)}$

${\displaystyle ={\frac {P({\text{awakening}}=1\cap X=1)}{P({\text{awakening}}=1\cup {\text{awakening}}=2)}}}$

${\displaystyle ={\frac {1/3}{1}}=1/3}$.

Analogous to Bayesian predictive probabilities, Sleeping Beauty could calculate frequentist predictive p-values and prediction confidence levels which describe the long-run performance of the prediction based on observed data without conditioning on parameter values. The predictive p-value is the probability of the discrepancy between the estimated result (prediction) and the hypothesized result or something more extreme. In this setting Sleeping Beauty's knowledge of the coin being fair is equivalent to having already flipped the coin a near infinite number of times (the estimate has no sampling variability). Therefore, the predictive p-value testing the hypothesis that the coin landed heads is equivalent to the limiting proportion of heads over repeated experiments where Sleeping Beauty is woken twice on tails and once on heads, ${\displaystyle P(X=1|{\text{awakening}}=1\cup {\text{awakening}}=2)=1/3}$. We are considering each prediction opportunity as an experiment and an opportunity to make a type I error, not each coin flip. Upon every awakening Sleeping Beauty will predict the coin landed tails, incorrectly rejecting the null hypothesis 33% of the time in repeated testing. Every time she is woken and tests the hypothesis that the coin landed heads she is ${\displaystyle 67\%}$ confident in the alternative, which is tails.

If asked for the limiting proportion of heads, the correct answer is ${\displaystyle 1/2}$ regardless of paradigm. If asked for the frequentist predictive p-value testing the hypothesis that the coin landed heads, the answer is ${\displaystyle 1/3}$. If asked for the updated Bayesian belief of the subject whether the coin landed heads, the answer is ${\displaystyle 1/3}$.

Nick Bostrom argues that the thirder position is implied by the Self-Indication Assumption.

Credence about what precedes awakenings is a core question in connection with the anthropic principle.

• Doomsday argument
• Bayesian probability
• Monty Hall problem

• Arntzenius, F (2002). "Reflections on Sleeping Beauty". Analysis. 62 (1): 53–62. doi:10.1093/analys/62.1.53. JSTOR 3329069.
• Bostrom, Nick (2002-07-12). Anthropic Bias. Routledge (UK). pp. 195–96. ISBN 978-0-415-93858-7.
• Bradley, D (2003). "Sleeping Beauty: a note on Dorr's argument for 1/3". Analysis. 63 (3): 266–268. doi:10.1093/analys/63.3.266. JSTOR 3329324.
• Bruce, Colin (2004-12-21). Schrodinger's Rabbits: Entering the Many Worlds of Quantum. Joseph Henry Press. pp. 193–96. ISBN 978-0-309-09051-3.
• Dorr, C (2002). "Sleeping Beauty: in Defence of Elga". Analysis. 62 (4): 292–296. doi:10.1093/analys/62.4.292. JSTOR 3328920.
• Elga, A. (2000). "Self-locating Belief and the Sleeping Beauty Problem". Analysis. 60 (2): 143–147. doi:10.1093/analys/60.2.143. JSTOR 3329167.
• Lewis, D. (2001). "Sleeping Beauty: reply to Elga" (PDF). Analysis. 61 (3): 171–76. doi:10.1093/analys/61.3.171. JSTOR 3329230.
• Meacham, C. J. (2008). "Sleeping beauty and the dynamics of de se beliefs". Philosophical Studies. 138 (2): 245–269. CiteSeerX 10.1.1.517.4904. doi:10.1007/s11098-006-9036-1. JSTOR 40208872. S2CID 26902640.
• Monton, B. (2002). "Sleeping Beauty and the Forgetful Bayesian". Analysis. 62 (1): 47–53. doi:10.1093/analys/62.1.47. JSTOR 3329068.
• Neal, R. (2006). Puzzles of Anthropic Reasoning Resolved Using Full Non-indexical Conditioning, preprint
• Rosenthal, J.S. (2009). "A mathematical analysis of the Sleeping Beauty problem". The Mathematical Intelligencer. 31 (3): 32–37. CiteSeerX 10.1.1.151.2326. doi:10.1007/s00283-009-9060-z. S2CID 14152244.
• Titelbaum, M. (2013). Quitting Certainties, 210–229, 233–237, 241–249, 250, 276–277
• Zuboff, A. (1990). "One Self: The Logic of Experience". Inquiry. 33 (1): 39–68. doi:10.1080/00201749008602210.

• Terry Horgan: Sleeping Beauty Awakened: New Odds at the Dawn of the New Day (review paper with references)
• Franceschi, Paul. "A Two-Sided Ontological Solution to the Sleeping Beauty Problem" (PDF).
• 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)
• Twoplustwo thread discussing the sleeping beauty problem in depth