If we accept this, then clearly 1⁄3 is correct. If we run this experiment multiple times and Beauty guessed 1⁄3 for heads, then we’d find heads actually came up 1⁄3 of the times she said “1/3”. Therefore, a well-calibrated Beauty guesses “1/3″.
On the other hand...
Intuitively, a well-calibrated person A should assign a probability of P% to X iff X happens on P% of the occasions where A assigned a P% probability to X.
Here we’re still left with “occasions”. Should a well-calibrated person be right half of the times they are asked, or about half of the events? If (on many trials) Beauty guesses “tails” every time, then she’s correct 2⁄3 of the times she’s asked. However, she’s correct 1⁄2 of the times that the coin is flipped.
If I ask you for the probability of ‘heads’ on a fair coin, you’ll come up with something like ‘1/2’. If I ask you a million times before flipping, flip once, and it comes up tails, and then ask you once more before flipping, flip once, and it comes up heads, then you should not count that as a million cases of ‘tails’ being the correct answer and one of ‘heads’, even though a guess of ‘tails’ would have made you correct on a million occasions of being asked the question.
“What is your credence now for the proposition that our coin landed heads?”
No mention of “occasions”. Your comment doesn’t seem to be addressing that question, but some other ones, which are not mentioned in the problem description.
This explains why you can defend the “wrong” answer: you are not addressing the original question.
I did not claim that the problem statement used the word “occasions”.
Beauty should answer whatever probability she would answer if she was well-calibrated. So does a well-calibrated Beauty answer ‘1/2’ or 1⁄3′? Does Laplace let her into Heaven or not?
By the way, do you happen to remember the name or location of the article in which Eliezer proposed the idea of being graded for your beliefs (by Laplace or whoever), by something like cross-entropy or K-L divergence, such that if you ever said about something true that it had probability 0, you’d be infinitely wrong?
On the other hand...
Here we’re still left with “occasions”. Should a well-calibrated person be right half of the times they are asked, or about half of the events? If (on many trials) Beauty guesses “tails” every time, then she’s correct 2⁄3 of the times she’s asked. However, she’s correct 1⁄2 of the times that the coin is flipped.
If I ask you for the probability of ‘heads’ on a fair coin, you’ll come up with something like ‘1/2’. If I ask you a million times before flipping, flip once, and it comes up tails, and then ask you once more before flipping, flip once, and it comes up heads, then you should not count that as a million cases of ‘tails’ being the correct answer and one of ‘heads’, even though a guess of ‘tails’ would have made you correct on a million occasions of being asked the question.
Well, the question was:
“What is your credence now for the proposition that our coin landed heads?”
No mention of “occasions”. Your comment doesn’t seem to be addressing that question, but some other ones, which are not mentioned in the problem description.
This explains why you can defend the “wrong” answer: you are not addressing the original question.
I did not claim that the problem statement used the word “occasions”.
Beauty should answer whatever probability she would answer if she was well-calibrated. So does a well-calibrated Beauty answer ‘1/2’ or 1⁄3′? Does Laplace let her into Heaven or not?
By the way, do you happen to remember the name or location of the article in which Eliezer proposed the idea of being graded for your beliefs (by Laplace or whoever), by something like cross-entropy or K-L divergence, such that if you ever said about something true that it had probability 0, you’d be infinitely wrong?
A Technical Explanation of Technical Explanation
What Nick said. Laplace is also mentioned jokingly in a different context in An Intuitive Explanation of Bayes’ Theorem.
Well, 1⁄3. I thought you were supposed to be defending the plausibility of the “1/2” answer here—not asking others which answer is right.