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I'm surely outing myself as a mathlet here, but perhaps you have the energy to explain where I err. I fully accept that if you are forced to put 10 dollars on a bet as to whether the coin was heads or tails every time you are awakened, then betting tails every time is the best strategy, in that it will pay out the most in the long-run.
Where I draw issue is equating this with "belief". If this experiment was going to be run on me right now, I would precommit to the tails-every-time betting strategy, but I would know the coin has 50-50 odds, and waking would not change that. To me, it seems the optimal betting strategy is separate from belief. Because in deciding it is the correct move to bet tails every time, I don't sincerely believe the coin will come up tails every time, I've merely decided this is the best betting strategy. I see no real connection between betting strategy and genuine belief.
Now where it is odd to me is that if you repeated the experiment on me 100 times, where 50 runs would be heads and 50 runs would be tails, then asked me while I was awoken what the odds I truly believe are, I would have no problem saying I think there is a 2/3 chance that I am in a tails experiment vs in a heads experiment. Why should one single experiment feel different and change that? I'm not entirely sure.
Hmm, there may be some misunderstanding about the term "belief" here (or "credence" from Wikipedia, or "confidence", all of which can kind of be used interchangeably)? You don't "believe" that the coin was tails (or heads). After awakening, what you believe is that there's a 2/3 chance that it was tails. Which, as you said, matches with your observations if you repeat the experiment 100 times, indicating that your belief is well-calibrated.
Wouldn't you have the same issue with "belief" without the whole experiment setup, if I just flipped a coin behind my back? Isn't it reasonable to say that you "believe" the coin has a 50-50 chance of being heads, if you can't see it?
Rationalists like to make probabilistic predictions for events all the time (which I sure hope reflects what they "believe"). If you read astralcodexten, he'll often post predictions with probabilities attached, and he considers his beliefs well-matched with the real world not by getting everything right, but by getting 9/10 of the 90% predictions right, 8/10 of the 80% predictions right, etc.
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