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dspeyer comments on Probability, knowledge, and meta-probability - Less Wrong
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The term "metaprobability" strikes me as adding confusion. The two layers are not the same thing applied to itself, but are in fact different questions. "What fraction of the time does this box pay out?" is a different question from "Is this box going to pay out on the next coin?".
Often it takes a lot of questions to fully describe a situation. Using the term "probability" for all of them hides the distinction.
But it is - you're answering the question "what is the probability that this box will pay out next time", and "what is the probability that my probability assignment was correct?"
What does it mean for a probability assignment to be correct, as opposed to well-calibrated? Reality is or is not.
I mostly meant well calibrated, but...
There is something-like-correctness in that, given the evidence available to you, there is a correct way to update your prior. That is strictly not a fact about your posterior, but I think it's a legitimate thing to talk about in terms of 'correctness'.