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Douglas_Knight comments on Which parts of philosophy are worth studying from a pragmatic perspective? - Less Wrong
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I agree that there are epistemological problems in the foundations, but they seem to me mild enough to refute frequentism. I'm not really sure what frequentism is, though (other than the position that one should not speak of the probability of a hypothesis). Can you spell out what you think the coherent frequentist position is? I won't hold it against you or frequentism if you say no.
(If one just speaks of beliefs, maybe there is a coherent frequentist position that evades Cox's theorem, but frequentists hold that we make decisions [ETA: a metaphysical but not epistemological assumption]; and this should be enough to force probabilistic beliefs.)
I'm not sure either, as I've confessed before, but here's how it seems to me: whereas Bayesians view beliefs as "thermometer readings" that go up and down as new information comes in, frequentists view beliefs as "certificates" that have to be "earned" through specific rituals, and are subject to periodic renewal.
Since (in view of results such as Cox's theorem) one can't really deny that probability theory is the appropriate mathematical model for belief if there is any appropriate model at all, it seems that what frequentists actually object to, deep down, is the idea of personal belief as the object of formalization. They are only comfortable with ("official") belief-formation as a social process. In their view (so I propose), you are only allowed to make probabilistic claims after you perform a well-defined experiment of some kind (specifically, of the kind in which relative frequencies of the events you're interested in can be observed).
See Science Doesn't Trust Your Rationality.
Frequentists don't have a defined view of beliefs, they have a defined view of probability, namely that probability statements about events are statements about the limit of that event's frequency in an arbitrary number of trials. The Bayesian position is that probability can be understood as degree of belief. The Bayesian epistemological position is that all beliefs ought to be understood as probability statements. There is no Frequentist epistemology, a frequentist could hold any epistemology they wanted more or less.
I understand that this is the standard story, but don't find it satisfactory; hence was presenting my own "deeper" explanation (which I still regard as tentative).
After all, based only on what you've said, a frequentist could also be a Bayesian! This doesn't seem right, or at least doesn't seem to account for the fact that there seems to be a controversy between "frequentists" on the one hand, and "Bayesians" on the other.
I don't see how a frequentist could Bayesian, to hold a Bayesian epistemology one would also have to adhere to a Bayesian theory of probability. Since frequentists don't hold such a theory that can not hold Bayesian epistemology.
As per my original comment, someone could agree that probabilities represent beliefs but also hold that the content of ("legitimate") beliefs consists only of statements about frequencies. This would allow them to simultaneously hold a frequentist interpretation of probability and a Bayesian epistemology. (As you said, "There is no Frequentist epistemology, a frequentist could hold any epistemology they wanted more or less.")
I'm not a Frequentist (so I won't bother writing up a justification for the position), but non-Bayesians like R. A. Fisher, Jerzy Neyman, and Egon Pearson didn't just dogmatically refuse to accept the conclusion of a valid mathematical argument. They denied the truth of Bayesianism's epistemological premises (note: I disagree with their judgment in this case). Not one of them denied that Baye's Rule could be derived from the very definition of conditional probability (which is a straightforward consequence of the mathematics).
My comment was intended to point out that it takes more than standard probability theory and deduction to get you to Bayesianism. Additional premises (from outside of mathematics) must be present (at least implicitly).
That's a reasonable response to Will's first two comments, but [ETA: not] as a response to his third comment, mentioning Cox's theorem, or my comment, mentioning decision theory. I don't blame you for not knowing whether they had a coherent system of beliefs, but I do blame you for this non sequitur.
ETA: maybe that would be reasonable if you just substituted Cox for Bayes, but only if these frequentists explicitly rejected their contemporary Cox, rather than just ignored him.