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Emile comments on Conditioning on Observers - Less Wrong
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In that case, what does it even mean to say "my subjective probability for heads is 1/2"? Subjective probability is often described in terms of bettings - see here.
Seems to me this is mostly a quarrel of definitions, and that when you say "people who believe the answer is 1/3 pictured counts in a 3 by 1 contingency table, and applied the wrong theory to it.", you're being unfair. They're just using a different definition of "subjective probability"
Don't you think so?
Based on my interaction with people here, I think we all are talking about the same thing when it comes to subjective probability.
I agree that you can use betting to describe subjective probability, but there are a lot of possible ways to bet.
"Subjective probability" is a basic term in decision theory and economics, though. If you want to roll your own metric, surely you should call it something else - to avoid much confusion.
That is why I'd rather talk in terms of bets than subjective probability - they don't require precise technical definitions.