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VipulNaik comments on Knightian Uncertainty from a Bayesian perspective - Less Wrong
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"Jonah was looking at probability distributions over estimates of an unknown probability (such as the probability of a coin coming up heads)"
It sounds like you are just confusing epistemic probabilities with propensities, or frequencies. I.e, due to physics, the shape of the coin, and your style of flipping, a particular set of coin flips will have certain frequency properties that you can characterise by a bias parameter p, which you call "the probability of landing on heads". This is just a parameter of a stochastic model, not a degree of belief.
However, you can have a degree of belief about what p is no problem. So you are talking about your degree of belief that a set of coin flips has certain frequentist properties, i.e. your degree of belief in a particular model for the coin flips.
edit: I could add that GIVEN a stochastic model you then have degrees of belief about whether a given coin flip will result in heads. But this is a conditional probability: see my other comment in reply to Vanvier. This is not, however, "beliefs about beliefs". It is just standard Bayesian modelling.
I understand this, though I hadn't thought of it with such clear terminology. I think the point Jonah was making was that in many cases, people are talking about propensities/frequencies when they refer to probabilities. So it's not so much that Jonah or I are confusing epistemic probabilities with propensities/frequencies, it's that many people use the term "probability" to refer to the latter. With language used this way, the probability distribution for this model parameter can be called the "probability distribution of the probability estimate." If you reserve the term probability exclusive to epistemic probability (degree of belief) then this would constitute an abuse of language.
Sure, I don't want to suggest we only use the word 'probability' for epistemic probabilities (although the world might be a better place if we did...), only that if we use the word to mean different sorts of probabilities in the same sentence, or even whole body of text, without explicit clarification, then it is just asking for confusion.