Except one makes it seem like the stuff does exist and the other makes it seem like it doesn’t. If we interpret the law of large numbers as saying that after an infinite number of trials, the average value of that sequence of results will equal the expected value of the random variable, then any finite amount of evidence is not enough to be evidence of this interpretation, let alone to verify it. But if we interpret the law as saying that the more trials we add, the more closely the average result should hover around the expected value of the variable. That interpretation can be falsified and evidenced empirically using only finite observations.
Ok, I am a Bayesian, i.e., I use bayesianism over frequentism, and find frequentest methods rather silly. And I am at least what I would call a finite frequentist, i.e., I think komolgorov models finite frequency.
I am not here to say that Bayesianism is on equal ground with Frequentism, at all. If Bayes’s interpreted sentences can be empirically verified, and freqeuntist interpretations cannot be empirically verified, than this is to bayesianism’s favor. It means it is the more useful theory. But it is not grounds to use “is” where we should use “models” instead. It is not because bayesains need to be put on equal footing with frequentists that I propose this terminology in place of the copula; it is because rationalists should be clear, specially in philosophy. So just to be clear, we should use “model” instead of “is” because it is what is really going on; the concepts of Hofstadterish formalism and model theory, are the best way to understand how probability theory ends up telling us how to distribute beliefs. The relation between subjective degree of belief, and probability theory, is clearly not identity, or the copula.
Subjective degrees of belief are a part of your cognition.
Probability theory is a repeatable process consisting in shuffling squiggles on paper, or some other medium.
These are not identical.
Q.e.d.
We might call this the “paper projection fallacy”. Where you project some pattern in squiggles on a piece of paper into your mind. Analogous to the “mind projection fallacy”.
Bayesian probability theory is the mathematical formulation representing ideal reasoning under uncertainty
The squiggles on the paper are our representation of this probability—they are “probability”, not probability, if you like.
no, probability, or I assume you really mean rationality, is the void
Bayes is just playing with squiggles on paper. If when you interpreted bayes, you found some claim, which seemed to not work, you would have to abandon bayes, or be irrational.
The squiggles on the paper are our representation of this probability.
What probability where? If you start by saying degree of belief is probability, and then show that degreeo f belief is probability, I am not impressed. You can call them “the representation” instead of calling them “the theory” if you want. And you can use “is” instead of model if you want. And you can even use “probability” instead of “degree of belief”, though I suspect that may all get rather confusing quickly. But do realize that every reason you give for saying that probability is degree of belief, a frequentest can give for saying that probability is frequency.
“Probability” is a really stupid noun, kind of like “red-hood”, or “emergence”. Notice how in the actual theory, we only ever talk about the probability of something. “Probability” is a function, not an object. Ask yourself: “what IS probability?” really probe, and you’ll find that that is a stupid question. The right question would have been, “what does probability return given an argument?” The answer is that it might return the rational degree of belief of a proposition, the frequency of a predicate out of a finite population, the frequency of a value out of an infinite amount of trials, the volume of a space, the area of a shape, or even the length of a line. All of these are consistent with the komologorov axioms.
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.”
Except one makes it seem like the stuff does exist and the other makes it seem like it doesn’t. If we interpret the law of large numbers as saying that after an infinite number of trials, the average value of that sequence of results will equal the expected value of the random variable, then any finite amount of evidence is not enough to be evidence of this interpretation, let alone to verify it. But if we interpret the law as saying that the more trials we add, the more closely the average result should hover around the expected value of the variable. That interpretation can be falsified and evidenced empirically using only finite observations.
.
Ok, I am a Bayesian, i.e., I use bayesianism over frequentism, and find frequentest methods rather silly. And I am at least what I would call a finite frequentist, i.e., I think komolgorov models finite frequency.
I am not here to say that Bayesianism is on equal ground with Frequentism, at all. If Bayes’s interpreted sentences can be empirically verified, and freqeuntist interpretations cannot be empirically verified, than this is to bayesianism’s favor. It means it is the more useful theory. But it is not grounds to use “is” where we should use “models” instead. It is not because bayesains need to be put on equal footing with frequentists that I propose this terminology in place of the copula; it is because rationalists should be clear, specially in philosophy. So just to be clear, we should use “model” instead of “is” because it is what is really going on; the concepts of Hofstadterish formalism and model theory, are the best way to understand how probability theory ends up telling us how to distribute beliefs. The relation between subjective degree of belief, and probability theory, is clearly not identity, or the copula.
Subjective degrees of belief are a part of your cognition. Probability theory is a repeatable process consisting in shuffling squiggles on paper, or some other medium. These are not identical. Q.e.d.
We might call this the “paper projection fallacy”. Where you project some pattern in squiggles on a piece of paper into your mind. Analogous to the “mind projection fallacy”.
.
no, probability, or I assume you really mean rationality, is the void
Bayes is just playing with squiggles on paper. If when you interpreted bayes, you found some claim, which seemed to not work, you would have to abandon bayes, or be irrational.
What probability where? If you start by saying degree of belief is probability, and then show that degreeo f belief is probability, I am not impressed. You can call them “the representation” instead of calling them “the theory” if you want. And you can use “is” instead of model if you want. And you can even use “probability” instead of “degree of belief”, though I suspect that may all get rather confusing quickly. But do realize that every reason you give for saying that probability is degree of belief, a frequentest can give for saying that probability is frequency.
“Probability” is a really stupid noun, kind of like “red-hood”, or “emergence”. Notice how in the actual theory, we only ever talk about the probability of something. “Probability” is a function, not an object. Ask yourself: “what IS probability?” really probe, and you’ll find that that is a stupid question. The right question would have been, “what does probability return given an argument?” The answer is that it might return the rational degree of belief of a proposition, the frequency of a predicate out of a finite population, the frequency of a value out of an infinite amount of trials, the volume of a space, the area of a shape, or even the length of a line. All of these are consistent with the komologorov axioms.
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.”