drmwc wrote:The key issue with the sleeping beauty problem is that SB has no way to differentiate Monday and Tuesday. This leads to some conterinuitive features.
Consider the statement "Given that the coin landed head, what is the probability that today is Tuesday?" Intuitively, the answer is 1/2. However, this leads to problems.
Wait a second. Which version is this about? If it is the one in which Sleeping Beauty is awakened on Tuesday if and only if the coin came up tails, then intuitively, the answer is 0.
Formally, we have the following statements we wish to be true:
Starting with what we wish to be true does not always work.
(1) P(Mon or Tue) = 1
(2) P(heads and Tue) = 0
(3) P(heads | Mon or Tue) = 1/2
(4) P(heads | Mon) = 1/2
(5) P(Mon | tails) = 1/2
I suppose you mean the conditions to be what Beauty knows.
We know that 1 is true from the set up of the experiment.
Similarly. we know that 2 is also true.
We want 3 to be true because we are told that the coin is fair (and presumably SB verified this before the experiment.) Also, SB gains no information from waking.
Of course Beauty gains information from waking. The probability that she wakes up on Monday is 1. The probability that she wakes up on Tuesday is 1/2. So when she wakes up, the odds are 2:1 that it is Monday. That's information.
Now, you may say that Beauty already had that figured out on Sunday. But on Sunday the probability that it was Monday was 0. It is easy to go astray with such statements as those about gaining or losing information when waking.
The question is what are the conditions of the probabilities. On Sunday, the probability was 1/2 that the coin would come up heads. On Monday and Tuesday, the probability that it had come up heads does not have to be the same as it was on Sunday, because the conditions are different.
We want 4 to be true, again because the coin is fair and SB gains no new information from waking on Monday.
What information Beauty gains from waking on Monday is not the question. The point is that Beauty always wakes up on Monday, regardless of the result of the coin toss. But on awakening, Beauty is not sure that it is Monday. She has to be told in order to get the "Mon" condition for 4.
We want 5 to be true for reasons such as the indfference principle.
That is one reason that probability is empirical.
However, the issue is that 1-5 above are inconsistent - they do not define a valid probability measure. (I've left the proof as an exercise to the reader.)
People who insist that the answer is 1/3 reject hypothesis 3 above, and insist that P(heads | Mon or Tue) = 1/3. This does give a consistent probability measure. However, changing the probabilty of heads when no new information is available is offensive to me, and feels hard to justify.
The "no new information" phrase is a trap. The right way to think about these things is in terms of the conditions of the probabilities. True, in one interpretation of "no new information", Sleeping Beauty gains no new information. But the conditions of the probabilities are different, as indicated above. That is what is important.
The neatest resolution is that 5) above has no defined probability. If, at the interview, SB is asked "What is the probability today is Monday?", her reply is: "I do not have sufficient information to answer".
The key issue with the question is the use of the word "today". This means the statement uses an indexical, and its truth value changes dependent on circumstances. Other indexicals include "I", "now", "this" etc.
Probability behaves badly when we attempt to assign probabilities to statements which include indexicals. Finding a replacement for the statement "Today is Monday" without fundamentally changing the problem is difficult - I have not seen it done.
For example, suppose an inspector arrvies either on Monday or Tuesday and attends the interview (if there is one). The day she arrives is chosen at random with probability 1/2, and is independent of the coin flip.
Now the symmetry of Monday and Tuesday collapses, and frequentists and Bayesians would both agree that probability of heads, given the inspector is present at the interview when asked, is 1/3. But this seems to violate the spirit of the oringinal experiment, where SB could not differentiate Monday and Tuesday.
Since Beauty cannot tell the difference between Monday and Tuesday, we should regard them as simply labels for different cases. I don't think that there is a problem with "today", meaning "this case", any more than there is a problem, if we draw two cards from a well shuffled standard deck and leave them face down, of then pointing to one card and asking what is the probability that
this card is a Jack.
But Monday and Tuesday are not symmetrical, anyway. Beauty just does not know which is which.
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Here is a variant that restores the symmetry between Monday and Tuesday as far as Beauty's awakening goes, but retains the asymmetry of the original problem. I'll skip the setup.
On Monday after Beauty awakes she is asked what is the probability that the coin came up heads. She is given the amnesia drug to make her forget the events of Monday, but she is allowed to awaken on Tuesday.
On Tuesday after Beauty awakes, if the coin came up tails she is asked what is the probability that it came up heads. If it came up heads, she is not asked anything about how the coin came up.
Upon awakening, what probability should Beauty assign to the coin coming up heads? Plainly, 1/2. When asked, what probability should she assign? Plainly -- I trust --, 1/3.
In this variation awakening does not affect her prior belief in the odds that the coin came up heads, but being asked (by the experimenters) does. She is more likely to be asked when the coin came up tails. When Beauty wakes up, Monday and Tuesday are symmetrical. It is the asking that breaks the symmetry.
If we eliminate the case in which Beauty wakes up but is not asked about how the coin came up, the symmetry is again destroyed.
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