“Now, it does seem to me that following the usual rules of probability theory … tells Sleeping Beauty to assign .5 credence to the proposition that the coin landed on heads.”
I think that your 1⁄3 answer is entirely consistent with the usual rules of probability theory, at least once a highly intuitive caveat is added: Given no new information, the odds do not change unless some old information has been lost. Usually, we implicitly assume that information is not lost so there’s no need for the caveat, but amnesia inducing drugs radically violate the assumption.
When sleeping beauty falls asleep on Sunday, she knows the day of the week, and that she’s been interviewed exactly zero times. When she wakes up, she doesn’t know the day or the number of times she’s been interviewed. This loss of information opens the door to the odds changing.
An easier way to think about the same problem in Bayesian terms is to compare her belief during an interviews to her belief afterward. On Wednesday, she wakes up and thinks that if there’s an interview, P(heads) = 1⁄3. But after realizing that there is no interview, she gains the information that it’s Wednesday. From this, she infers that her experiences are no longer being skewed toward worlds where the result is tails by a factor of 2, so she updates her belief to P(heads) = 1⁄2.
“Now, it does seem to me that following the usual rules of probability theory … tells Sleeping Beauty to assign .5 credence to the proposition that the coin landed on heads.”
I think that your 1⁄3 answer is entirely consistent with the usual rules of probability theory, at least once a highly intuitive caveat is added: Given no new information, the odds do not change unless some old information has been lost. Usually, we implicitly assume that information is not lost so there’s no need for the caveat, but amnesia inducing drugs radically violate the assumption.
When sleeping beauty falls asleep on Sunday, she knows the day of the week, and that she’s been interviewed exactly zero times. When she wakes up, she doesn’t know the day or the number of times she’s been interviewed. This loss of information opens the door to the odds changing.
An easier way to think about the same problem in Bayesian terms is to compare her belief during an interviews to her belief afterward. On Wednesday, she wakes up and thinks that if there’s an interview, P(heads) = 1⁄3. But after realizing that there is no interview, she gains the information that it’s Wednesday. From this, she infers that her experiences are no longer being skewed toward worlds where the result is tails by a factor of 2, so she updates her belief to P(heads) = 1⁄2.