I think it’s not the change in perspective or subjective identity making a difference, but instead it’s a change in precisely which probability is being asked about. The Wikipedia page unhelpfully conflates the two changes.
It says that the experimenter must see a probability of 1⁄2 and Beauty must see a probability of 1⁄3, but that just ain’t so; there is nothing stopping Beauty from caring about the proportion of coin flips that turn out to be heads (which is 1⁄2), and there is nothing stopping the experimenter from caring about the proportion of wakings for which the coin is heads (which is 1⁄3). You can change which probability you care about without changing your subjective identity and vice versa.
Let’s say I’m Sleeping Beauty. I would interpret the question as being about my estimate of a probability (‘credence’) associated with a coin-flipping process. Having interpreted the question as being about that process, I would answer 1⁄2 - who I am would have nothing to do with the question’s correct answer, since who I am has no effect on the simple process of flipping a fair coin and I am given no new information after the coin flip about the coin’s state.
“What is your credence now for the proposition that our coin landed heads?”
That’s fairly clearly the PROBABILITY NOW of the coin having landed heads—and not the PROPORTION that turn out AT SOME POINT IN THE FUTURE to have landed heads.
Perspective can make a difference—because different observers have different levels of knowledge about the situation. In this case, Beauty doesn’t know whether it is Tuesday or not—but she does know that if she is being asked on Tuesday, then the coin came down tails—and p(heads) is about 0.
In the original problem post, Beauty is asked a specific question, though
It’s not specific enough. It only asks for Beauty’s credence of a coin landing heads—it doesn’t tell her to choose between the credence of a coin landing heads given that it is flipped and the credence of a coin landing heads given a single waking. The fact that it’s Beauty being asked does not, in and of itself, mean the question must be asking the latter probability. It is wholly reasonable for Beauty to interpret the question as being about a coin-flipping process for which the associated probability is 1⁄2.
That’s fairly clearly the PROBABILITY NOW of the coin having landed heads—and not the PROPORTION that turn out AT SOME POINT IN THE FUTURE to have landed heads.
The addition of the word ‘now’ doesn’t magically ban you from considering a probability as a limiting relative frequency.
Perspective can make a difference—because different observers have different levels of knowledge about the situation. In this case, Beauty doesn’t know whether it is Tuesday or not
Agree.
- but she does know that if she is being asked on Tuesday, then the coin came down tails—and p(heads) is about 0.
It’s not clear to me how this conditional can be informative from Beauty’s perspective, as she doesn’t know whether it’s Tuesday or not. The only new knowledge she gets is that she’s woken up; but she has an equal probability (i.e. 1) of getting evidence of waking up if the coin’s heads or if the coin’s tails. So Beauty has no more knowledge than she did on Sunday.
She has LESS knowledge than she had on Sunday in one critical area—because now she doesn’t know what day of the week it is. She may not have learned much—but she has definitely forgotten something—and forgetting things changes your estimates of their liklihood just as much as learning about them does.
She has LESS knowledge than she had on Sunday in one critical area—because now she doesn’t know what day of the week it is. She may not have learned much—but she has definitely forgotten something -
That’s true.
and forgetting things changes your estimates of their liklihood just as much as learning about them does.
I’m not as sure about this. It’s not clear to me how it changes the likelihoods if I sketch Beauty’s situation at time 1 and time 2 as
A coin will be flipped and I will be woken up on Monday, and perhaps Tuesday. It is Sunday.
I have been woken up, so a coin has been flipped. It is Monday or Tuesday but I do not know which.
as opposed to just
A coin will be flipped and I will be woken up on Monday, and perhaps Tuesday.
I have been woken up, so a coin has been flipped. It is Monday or Tuesday but I do not know which.
(Edit to clarify—the 2nd pair of statements is meant to represent roughly how I was thinking about the setup when writing my earlier comment. That is, it’s evident that I didn’t account for Beauty forgetting what day of the week it is in the way timtyler expected, but at the same time I don’t believe that made any material difference.)
I think it’s not the change in perspective or subjective identity making a difference, but instead it’s a change in precisely which probability is being asked about. The Wikipedia page unhelpfully conflates the two changes.
It says that the experimenter must see a probability of 1⁄2 and Beauty must see a probability of 1⁄3, but that just ain’t so; there is nothing stopping Beauty from caring about the proportion of coin flips that turn out to be heads (which is 1⁄2), and there is nothing stopping the experimenter from caring about the proportion of wakings for which the coin is heads (which is 1⁄3). You can change which probability you care about without changing your subjective identity and vice versa.
Let’s say I’m Sleeping Beauty. I would interpret the question as being about my estimate of a probability (‘credence’) associated with a coin-flipping process. Having interpreted the question as being about that process, I would answer 1⁄2 - who I am would have nothing to do with the question’s correct answer, since who I am has no effect on the simple process of flipping a fair coin and I am given no new information after the coin flip about the coin’s state.
In the original problem post, Beauty is asked a specific question, though—namely:
“What is your credence now for the proposition that our coin landed heads?”
That’s fairly clearly the PROBABILITY NOW of the coin having landed heads—and not the PROPORTION that turn out AT SOME POINT IN THE FUTURE to have landed heads.
Perspective can make a difference—because different observers have different levels of knowledge about the situation. In this case, Beauty doesn’t know whether it is Tuesday or not—but she does know that if she is being asked on Tuesday, then the coin came down tails—and p(heads) is about 0.
It’s not specific enough. It only asks for Beauty’s credence of a coin landing heads—it doesn’t tell her to choose between the credence of a coin landing heads given that it is flipped and the credence of a coin landing heads given a single waking. The fact that it’s Beauty being asked does not, in and of itself, mean the question must be asking the latter probability. It is wholly reasonable for Beauty to interpret the question as being about a coin-flipping process for which the associated probability is 1⁄2.
The addition of the word ‘now’ doesn’t magically ban you from considering a probability as a limiting relative frequency.
Agree.
It’s not clear to me how this conditional can be informative from Beauty’s perspective, as she doesn’t know whether it’s Tuesday or not. The only new knowledge she gets is that she’s woken up; but she has an equal probability (i.e. 1) of getting evidence of waking up if the coin’s heads or if the coin’s tails. So Beauty has no more knowledge than she did on Sunday.
She has LESS knowledge than she had on Sunday in one critical area—because now she doesn’t know what day of the week it is. She may not have learned much—but she has definitely forgotten something—and forgetting things changes your estimates of their liklihood just as much as learning about them does.
That’s true.
I’m not as sure about this. It’s not clear to me how it changes the likelihoods if I sketch Beauty’s situation at time 1 and time 2 as
A coin will be flipped and I will be woken up on Monday, and perhaps Tuesday. It is Sunday.
I have been woken up, so a coin has been flipped. It is Monday or Tuesday but I do not know which.
as opposed to just
A coin will be flipped and I will be woken up on Monday, and perhaps Tuesday.
I have been woken up, so a coin has been flipped. It is Monday or Tuesday but I do not know which.
(Edit to clarify—the 2nd pair of statements is meant to represent roughly how I was thinking about the setup when writing my earlier comment. That is, it’s evident that I didn’t account for Beauty forgetting what day of the week it is in the way timtyler expected, but at the same time I don’t believe that made any material difference.)