I’m not asking about teaching or textbook-writing; my question is a tangent to that discussion. Smoofra invoked a notion of “impact”; I’m trying to determine how smoofra rates Cox’s theorem on that scale.
I know of a number of paths to the Bayesian approach:
Dutch book arguments (coherence of bets)
more elaborate decision theory arguments in the same vein (coherence of decisions under uncertainty)
the complete class theorem (the set of decision rules “admissible” (non-dominated) in a frequentist sense is precisely the set of Bayes decision rules)
de Finetti’s theorem (exchangeability of observables implies the existence of a prior and posterior for a parameter as a mathematical fact)
Cox’s theorem (a representation of plausibility as a single real number consistent with Boolean logic must be isomorphic to probability)
My question is about which justifications smoofra knows (in particular, have I missed any?) and what impact each of them has.
FWIW, in high school my very first introduction to integrals used the high-school version of Riemann integration with a simple definite integral that was solved with algebra and the notion of the limit of a sequence. Once it was demonstrated that the area under the curve was the anti-derivative in that case, we got the statement of the Fundamental Theorem of Calculus and drilling in anti-derivatives and integration by parts etc.
I’m not asking about teaching or textbook-writing; my question is a tangent to that discussion. Smoofra invoked a notion of “impact”; I’m trying to determine how smoofra rates Cox’s theorem on that scale.
I know of a number of paths to the Bayesian approach:
Dutch book arguments (coherence of bets)
more elaborate decision theory arguments in the same vein (coherence of decisions under uncertainty)
the complete class theorem (the set of decision rules “admissible” (non-dominated) in a frequentist sense is precisely the set of Bayes decision rules)
de Finetti’s theorem (exchangeability of observables implies the existence of a prior and posterior for a parameter as a mathematical fact)
Cox’s theorem (a representation of plausibility as a single real number consistent with Boolean logic must be isomorphic to probability)
My question is about which justifications smoofra knows (in particular, have I missed any?) and what impact each of them has.
FWIW, in high school my very first introduction to integrals used the high-school version of Riemann integration with a simple definite integral that was solved with algebra and the notion of the limit of a sequence. Once it was demonstrated that the area under the curve was the anti-derivative in that case, we got the statement of the Fundamental Theorem of Calculus and drilling in anti-derivatives and integration by parts etc.