Now, can’t I be a philosophical frequentest and a subjective bayesian? Just because probability theory models subjective beliefs does not mean that it doesn’t model frequencies; in fact, if some body told me that bayes doesn’t model frequencies I’m pretty sure I could prove them wrong much more easily than someone who said that probabilities don’t model degrees of belief.
But there is no contradiction in saying that the komolgorov probability function models both degrees of beliefs and actual frequencies.
(edit)
In fact it seems to me that komolgorov plainly does model frequency since it models odds, and odds model frequencies by a simple conversion. In fact, degrees of belief seem to model frequency as well. Using the thought experiment of frequencies of worlds you think you might find yourself in makes it simple to see how at least some degrees of belief can be seen as frequencies in and of themselves. In this thought experiment we treat probability as a measure of the worlds you think you might find yourself in; if you think that “there are ten cards and eight of them are blue”, then in 4/5s of the the worlds where “there are ten cards and eight of them are blue” holds, “the top card is blue” also holds. So you rightfully assign a 80% probability to the top card being blue.
Where the frequentest makes an error is in thinking that probabilities are then out there in the world treated as degrees of belief. What I mean by this is that they take the step from frequencies being in the world to uncertainties being in the world. This is a mistake, but I think that it is not central to the philosophical doctrine of frequent-ism. All that some frequentests claim is that probability models frequency and this is plainly true. And it is also true that there are frequencies in the world. Real frequencies independent of our minds. These are not probabilities, because there are no probabilities anywhere. Not even in minds.
Probability is not degree of subjective belief, probability is a class of automatized functions. These automized function model a great deal of things, measure theory, euclidean geometry, the constraints of rational beliefs, set cardinality, frequencies, etc. and the list can go on and on. Probability is a mathematical tool. And it is isomorphic to many important features of rationality and science, perhaps the most important being subjective degree of belief. But to argue that probability is subjective degree of belief just because it models degree of belief seems as silly to me as arguing that probability is frequency just because it models frequency. Why not the probability position of measure. Measure theory is isomorphic to probability. Why not say that probability is measure? Add that to the debating line.
I think the position to take towards probability is a properly Hofstadter-ish-ian formalism. Where the true statements about probability are simply the statements which are formed when you interpret the theorems of probability theory. Whatever else probability may be able to talk about truthfully it does so through isomorphism.
This is pretty close to my own position… Probability is strictly a mathematical concept (Kolmogorov axioms).
Real-world probability is anything that can be successfully modelled by the Kolmogorov axioms.
This applies to both betting probabilities (violate the axioms and you get Dutch-booked) and relative frequencies.
I’m a little bit puzzled by Eliezer’s view that probability is purely Bayesian, as he also believes in a “Big World”, and the relative frequency approach works extremely well in a Big World (as long as it is an infinitely-big world). Chancy events really do get repeated infinitely many times, the repetitions really are independent (because of large separation, and locality of physics), and the relative frequencies really are defined and really do converge to exactly what QM says the probabilities are. All works fine.
Also, there is a formal isomorphism between decoherent branches of a wave function (as applied to a single causal region) and spatially-separated causal regions in a multiverse. So you can, if you like, consider a single space time multiverse with an intuitive interpretation (other universes are just really far away) and forget about all the splitting. Bousso and Susskind have a nice recent paper about this: http://arxiv.org/abs/1105.3796
Now, can’t I be a philosophical frequentest and a subjective bayesian? Just because probability theory models subjective beliefs does not mean that it doesn’t model frequencies; in fact, if some body told me that bayes doesn’t model frequencies I’m pretty sure I could prove them wrong much more easily than someone who said that probabilities don’t model degrees of belief.
But there is no contradiction in saying that the komolgorov probability function models both degrees of beliefs and actual frequencies.
(edit)
In fact it seems to me that komolgorov plainly does model frequency since it models odds, and odds model frequencies by a simple conversion. In fact, degrees of belief seem to model frequency as well. Using the thought experiment of frequencies of worlds you think you might find yourself in makes it simple to see how at least some degrees of belief can be seen as frequencies in and of themselves. In this thought experiment we treat probability as a measure of the worlds you think you might find yourself in; if you think that “there are ten cards and eight of them are blue”, then in 4/5s of the the worlds where “there are ten cards and eight of them are blue” holds, “the top card is blue” also holds. So you rightfully assign a 80% probability to the top card being blue.
Where the frequentest makes an error is in thinking that probabilities are then out there in the world treated as degrees of belief. What I mean by this is that they take the step from frequencies being in the world to uncertainties being in the world. This is a mistake, but I think that it is not central to the philosophical doctrine of frequent-ism. All that some frequentests claim is that probability models frequency and this is plainly true. And it is also true that there are frequencies in the world. Real frequencies independent of our minds. These are not probabilities, because there are no probabilities anywhere. Not even in minds.
Probability is not degree of subjective belief, probability is a class of automatized functions. These automized function model a great deal of things, measure theory, euclidean geometry, the constraints of rational beliefs, set cardinality, frequencies, etc. and the list can go on and on. Probability is a mathematical tool. And it is isomorphic to many important features of rationality and science, perhaps the most important being subjective degree of belief. But to argue that probability is subjective degree of belief just because it models degree of belief seems as silly to me as arguing that probability is frequency just because it models frequency. Why not the probability position of measure. Measure theory is isomorphic to probability. Why not say that probability is measure? Add that to the debating line.
I think the position to take towards probability is a properly Hofstadter-ish-ian formalism. Where the true statements about probability are simply the statements which are formed when you interpret the theorems of probability theory. Whatever else probability may be able to talk about truthfully it does so through isomorphism.
This is pretty close to my own position… Probability is strictly a mathematical concept (Kolmogorov axioms). Real-world probability is anything that can be successfully modelled by the Kolmogorov axioms. This applies to both betting probabilities (violate the axioms and you get Dutch-booked) and relative frequencies.
I’m a little bit puzzled by Eliezer’s view that probability is purely Bayesian, as he also believes in a “Big World”, and the relative frequency approach works extremely well in a Big World (as long as it is an infinitely-big world). Chancy events really do get repeated infinitely many times, the repetitions really are independent (because of large separation, and locality of physics), and the relative frequencies really are defined and really do converge to exactly what QM says the probabilities are. All works fine.
Also, there is a formal isomorphism between decoherent branches of a wave function (as applied to a single causal region) and spatially-separated causal regions in a multiverse. So you can, if you like, consider a single space time multiverse with an intuitive interpretation (other universes are just really far away) and forget about all the splitting. Bousso and Susskind have a nice recent paper about this: http://arxiv.org/abs/1105.3796