I completely agree with what you said, but notice that this inference works the same no matter the actual facts about the day that you wake up; all hypothetical copies get the same answer. And if you know with certainty the outcome of a future computation, you should just update on it right away… which implies that the coin is unfair before you ever flip it, and that you can manipulate the coin probabilities by just precommiting to particular setups of the problem (n wake ups for heads, m wake ups for tails).
Given the above formulation, the inference is not the same. P(H) = 1⁄2, but after information about W or not W is given, then P(H | W) = 1⁄3 or P(H | not W) = 1. The math doesn’t care, you just aren’t awake to perform your update process. When precommitting, you are not manipulating P(H), you are manipulating P(H |W) by changing W, so there’s no issue.
P(A) = 1⁄2, P(B) = 0 is still the only way I can see to get P(H | W) = 1⁄2. In which case, I can’t find any non-artificial framing for why Heads Tuesday does not exist (and Heads Monday exists twice as much as Tails Monday).
I completely agree with what you said, but notice that this inference works the same no matter the actual facts about the day that you wake up; all hypothetical copies get the same answer. And if you know with certainty the outcome of a future computation, you should just update on it right away… which implies that the coin is unfair before you ever flip it, and that you can manipulate the coin probabilities by just precommiting to particular setups of the problem (n wake ups for heads, m wake ups for tails).
Given the above formulation, the inference is not the same. P(H) = 1⁄2, but after information about W or not W is given, then P(H | W) = 1⁄3 or P(H | not W) = 1. The math doesn’t care, you just aren’t awake to perform your update process. When precommitting, you are not manipulating P(H), you are manipulating P(H |W) by changing W, so there’s no issue.
P(A) = 1⁄2, P(B) = 0 is still the only way I can see to get P(H | W) = 1⁄2. In which case, I can’t find any non-artificial framing for why Heads Tuesday does not exist (and Heads Monday exists twice as much as Tails Monday).