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Rain and fog make for dreary DC forecast; wintry mix possible Monday. Guice, Redskins hold on to beat Panthers If the coin landed heads, she is only awakened on Monday, just like in the original experiment. Following the original thirder argument, on each awakening, SB can now be almost certain that the coin landed tails: instead of just three, there are one million and one possible outcomes, and just one of them implies heads.
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Suppose that we change the experiment such that if the coin landed heads, SB will not be awakened at all. If the coin landed tails, SB will be awakened at least once. Still, invoking the thirder argument in this context spells trouble of a different sort. Consider the Presumptuous Philosopher : the Presumptuous Philosopher deems any cosmological model that predicts that our universe contains a trillion trillion conscious observers a trillion times more likely than any model that predicts only a trillion of such observers.
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Consequently, he or she could — in principle — discount the majority of theories of physical reality by an armchair argument. You are awakened only once on heads, and a million times on tails. Even if the coin had such a large bias favoring heads, the thirder position tells us that it is still far more likely that the coin landed tails when you awake. The interviewer, an evil scientist, will kill you on the spot if you give the wrong answer. I will come back to this variation in the concluding section.
M odifying the number of awakenings is one way to generalize Sleeping Beauty.
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Another is to assume an unfair coin, and take the original halfer and thirder arguments along the ride in order to test their robustness against such generalization. We will once more assume the original number of awakenings: One awakening if the coin landed heads, two awakenings if it landed tails. SB knows the value of p ; alternatively she might be given the opportunity to estimate p herself by tossing the coin a lot of times before the experiment is conducted.
However, both positions disagree on what conditioning on Monday implies:. Halfer position. Thirder position. In order for us to generalize the thirder position to biased coins, we need to adjust the original arguments. One method to choose this parameter could be the principle of maximum entropy: without further information, we should choose the distribution closest to being uniform.
As a function of the parameter q , given p , the entropy of the above distribution can be computed as.
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This analysis leads to the following conclusion:. This state transition diagram should be read as follows: When SB is awake on Monday, with probability p her next awakening will, again, be on a Monday: there will be no interview on a Tuesday when the coin lands heads, and the experiment will be repeated. With probability 1 - p , her next awakening will be on a Tuesday before the experiment will be repeated. Suppose that the experiment has already been repeated a lot of times. In this model, the halfer and thirder solutions take on a remarkable characterization:. Markovian halfer.
The probability that the next transition that is either heads or tails will be a heads transition, is given by p :. Markovian thirder. While we are still confronted with the same distinct solutions when approaching the problem by modeling it as a Markov chain, I feel that it leads to a clearer picture as to the interpretation of these solutions. However, in this context, I will give an argument for the halfer position, by means of a variation of the repeated Sleeping Beauty experiment as follows. In the original formulation, SB is fully aware of the mechanism that generates the data but she has no access to actual data.
Now suppose that SB is told with each awakening whether the coin landed heads, and she is allowed to keep track of the number of awakenings. However, this time she is unaware of the elaborate procedure with which these data have been generated. However, she can also learn that certain features never seem to appear, such as HTH. Therefore, SB is given all the data, and she is not even aware of any particular reference class with respect to which she should compute frequencies.
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When she is given the details of the experiments after the fact, she will find the superior theory confirmed. Furthermore, there are two ways to marry both halfer and thirder views by providing a model that generates the data unobserved by SB when executing the experiment repeatedly:. Personally, I have changed my view from a thirder to a halfer while pondering the problem.
This can also be exemplified by different utilities:. However, I am convinced that the discussion does not end here. The Sleeping Beauty problem , video by Julia Galef. Piccione and A. In: Games and Economic Behavior 20 , pp. In: Analysis Guiasu and A. In: The Mathematical Intelligencer 7. Reversibility and Stochastic Networks.