while also not believing in certain spooky things called “continuous random variables”, which don’t really fit into Cox’s Theorem very well, if I understood Jaynes correctly.
I found a partial answer to the question I asked in the sibling comment. By chance I happened to need to generate random chords of a circle covering the circle uniformly. In searching on the net for Jaynes’ solution I came across a few fragments of Jaynes’ views on infinity. In short, he insists on always regarding continuous situations as limits of finite ones (e.g as when the binomial distribution tends to the normal), which is unproblematic for all the mathematics he wants to do. That is how the real numbers are traditionally formalised anyway. All of analysis is left unscathed. His wider philosophical objections to such things as Cantor’s transfinite numbers can be ignored, since these play no role in statistics and probability anyway.
I don’t know about the technicalities regarding Cox’s Theorem, but I do notice a substantial number of papers arguing about exactly what hypotheses it requires or does not require, and other papers discussing counterexamples (even to the finite case). The Wikipedia article has a long list of references, and a general search shows more. 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 can google /”Cox’s theorem” review/ but it is difficult for me to judge where the results sit within current understanding, or indeed what the current understanding is.
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 found a partial answer to the question I asked in the sibling comment. By chance I happened to need to generate random chords of a circle covering the circle uniformly. In searching on the net for Jaynes’ solution I came across a few fragments of Jaynes’ views on infinity. In short, he insists on always regarding continuous situations as limits of finite ones (e.g as when the binomial distribution tends to the normal), which is unproblematic for all the mathematics he wants to do. That is how the real numbers are traditionally formalised anyway. All of analysis is left unscathed. His wider philosophical objections to such things as Cantor’s transfinite numbers can be ignored, since these play no role in statistics and probability anyway.
I don’t know about the technicalities regarding Cox’s Theorem, but I do notice a substantial number of papers arguing about exactly what hypotheses it requires or does not require, and other papers discussing counterexamples (even to the finite case). The Wikipedia article has a long list of references, and a general search shows more. 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 can google /”Cox’s theorem” review/ but it is difficult for me to judge where the results sit within current understanding, or indeed what the current understanding is.
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.