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
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