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neq1 comments on Conditioning on Observers - Less Wrong
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The point of the PSB problem is that the approach you've just outlined is indefensible.
You agree that for each single constant k_i P(H|W) = 1/21. Uncertainty over which constant k_i is used does not alter this.
So if I run PSB 20 times, you would assert in each run that P(H|W) = 1/21. So now I simply keep you sedated between experiments. Statistically, 20 runs yields you SB, and each time you answered with 1/21 as your credence. Does this not faze you at all?
You have a scenario A where you assert foo with credence P, and scenario B where you also assert foo with credence P, yet if I put you in scenario A and then scenario B, keeping you sedated in the meantime, you do not assert foo with credence P...
You just changed the problem. If you wake me up between runs of PSB, then P(H|W)=1/21 each time. If not, I have different information to condition on.
No; between sedation and amnesia you know nothing but the fact that you've been woken up, and that 20 runs of this experiment are to be performed.
Why would an earlier independent trial have any impact on you or your credences, when you can neither remember it nor be influenced by it?
I don't know. It's a much more complicated problem, because you have 20 coin flips (if I understand the problem correctly). I haven't taken the time to work through the math yet. It's not obvious to me, though, why this corresponds to the sleeping beauty problem. In fact, it seems pretty clear that it doesn't.
The reason it corresponds to Sleeping Beauty is that in the limit of a large number of trials, we can consider blocks of 20 trials where heads was the flip and all values of the die roll occurred, and similar blocks for tails, and have some epsilon proportion left over. (WLLN)
Each of those blocks corresponds to Sleeping Beauty under heads/tails.