This assumes an “expected value” which could only be known by some other means, i.e. accepting the Bayesian notion of probability as subjective degrees of belief, or supposing an infinite number of trials. Such a definition of frequentism begs the question.
Well it is actually in the bartender’s premise that the coin is biased, so they both know that whatever heads/trials hovers around as trials rises, it is not 1⁄2.
But assuming they didn’t have that premise, what could the frequentist do, without requiring non-empirically verifiable claims as assumptions?
Only thing i can think of: He/she could resort to ranges. Never actually defining the probability of heads, just determining the probability with which the actual probability i.e. frequency of heads, is within a given range. There is some ideal actual frequency, which would be the outcome given infinitely many trials, but you can only find a range within which it is, and it would require infinite amounts of evidence to constrain heads/trials to a point; and we don’t have that kind of time. Bayes can be extended to ranges of probability trivially. THis would make it so that finite observables act as evidence for some hypotheses which include the term “infinity”. But it wouldn’t justify the whole of frequentest methodology.
But again, even if the frequentest interpretation fails in ways which the bayesian interpretation does not, this is not evidence of probability being degree of belief. It is evidence of probability modeling degree of belief, and of Bayesianism having sounder ontological commitments than frequentism. This would not surprise me.
Infinite frequency is not real. But our intuitions about it are real. Komolgorov may then be said to model actual finite frequencies, and our intuitions about infinite frequencies which are finitely and axiomatically describable. Let us not forget that there are not circles or squares anywhere either, but we should still hold that you can’t square the circle. Not all models have to be out there, some may be in here Frequentism requires infinite frequencies for its interpretation to be true, which don’t exist. The subjective bayes interpretation of bayes does not require anything that really doesn’t exist (though degrees of belief are plenty mysterious). This is a good reason to be a subjective Bayesian, and not a frequentest, which I was not aware of consciously, but it is not a good reason to stop being a formalist.
Who cares if frequentists, or non-LW bayesians, use the copula like a bunch of sillies, even after G.E.B. is published. We LWers, should use “identity” if we are claiming identity, and “modeling” if we are claiming a model. But realistically, the claim that “Probability theory models rational belief systems.” seems much more defensible, concrete, and useful, than the claim that “Probability is degree of belief.”
Now for this part:
Well it is actually in the bartender’s premise that the coin is biased, so they both know that whatever heads/trials hovers around as trials rises, it is not 1⁄2.
But assuming they didn’t have that premise, what could the frequentist do, without requiring non-empirically verifiable claims as assumptions?
Only thing i can think of: He/she could resort to ranges. Never actually defining the probability of heads, just determining the probability with which the actual probability i.e. frequency of heads, is within a given range. There is some ideal actual frequency, which would be the outcome given infinitely many trials, but you can only find a range within which it is, and it would require infinite amounts of evidence to constrain heads/trials to a point; and we don’t have that kind of time. Bayes can be extended to ranges of probability trivially. THis would make it so that finite observables act as evidence for some hypotheses which include the term “infinity”. But it wouldn’t justify the whole of frequentest methodology.
But again, even if the frequentest interpretation fails in ways which the bayesian interpretation does not, this is not evidence of probability being degree of belief. It is evidence of probability modeling degree of belief, and of Bayesianism having sounder ontological commitments than frequentism. This would not surprise me.
Infinite frequency is not real. But our intuitions about it are real. Komolgorov may then be said to model actual finite frequencies, and our intuitions about infinite frequencies which are finitely and axiomatically describable. Let us not forget that there are not circles or squares anywhere either, but we should still hold that you can’t square the circle. Not all models have to be out there, some may be in here Frequentism requires infinite frequencies for its interpretation to be true, which don’t exist. The subjective bayes interpretation of bayes does not require anything that really doesn’t exist (though degrees of belief are plenty mysterious). This is a good reason to be a subjective Bayesian, and not a frequentest, which I was not aware of consciously, but it is not a good reason to stop being a formalist.
Who cares if frequentists, or non-LW bayesians, use the copula like a bunch of sillies, even after G.E.B. is published. We LWers, should use “identity” if we are claiming identity, and “modeling” if we are claiming a model. But realistically, the claim that “Probability theory models rational belief systems.” seems much more defensible, concrete, and useful, than the claim that “Probability is degree of belief.”