Has anyone written an up to date review of what Cox-style theorems are known to be sound and how well they suffice to found the mathematics of probability theory?
I don’t know. But I will say this: I am distrustful of a foundation which takes “propositions” to be primitive objects. If the Cox’s Theorem foundation for probability requires that we assume a first-order logic foundation of mathematics in general, in which propositions cannot be considered as instances of some larger class of things (as they can in, for personal favoritism, type theory), then I’m suspicious.
I’m also suspicious of how Cox’s Theorem is supposed to map up to continuous and non-finitary applications of probability—even discrete probability theory, as when dealing with probabilistic programming or the Solomonoff measure. In these circumstances we seem to need the measure-theoretic approach.
Further: if “the extension of classical logic to continuous degrees of plausibility” and “rational propensities to bet” and “measure theory in spaces of normed measure” and “sampling frequencies in randomized conditional simulations of the world” all yield the same mathematical structure, then I think we’re looking at something deeper and more significant than any one of these presentations admits.
In fact, I’d go so far as to say there isn’t really a “Bayesian/Frequentist dichotomy” so much as a “Bayesian-Frequentist Isomorphism”, in the style of the Curry-Howard Isomorphism. Several things we thought were different are actually the same.
I don’t know. But I will say this: I am distrustful of a foundation which takes “propositions” to be primitive objects. If the Cox’s Theorem foundation for probability requires that we assume a first-order logic foundation of mathematics in general, in which propositions cannot be considered as instances of some larger class of things (as they can in, for personal favoritism, type theory), then I’m suspicious.
I’m also suspicious of how Cox’s Theorem is supposed to map up to continuous and non-finitary applications of probability—even discrete probability theory, as when dealing with probabilistic programming or the Solomonoff measure. In these circumstances we seem to need the measure-theoretic approach.
Further: if “the extension of classical logic to continuous degrees of plausibility” and “rational propensities to bet” and “measure theory in spaces of normed measure” and “sampling frequencies in randomized conditional simulations of the world” all yield the same mathematical structure, then I think we’re looking at something deeper and more significant than any one of these presentations admits.
In fact, I’d go so far as to say there isn’t really a “Bayesian/Frequentist dichotomy” so much as a “Bayesian-Frequentist Isomorphism”, in the style of the Curry-Howard Isomorphism. Several things we thought were different are actually the same.