My view is that writing a textbook on Bayesian statistics is very difficult because it is hard to order the material in a satisfactory way.
Here is why I’m wrong: when we teach calculus we teach differentiation. Then we say that integrals are important because they are areas under curves. Then we drill our pupils in the computation of integrals by finding anti-derivatives. It is only two or three years later that we introduce the Riemann integral in order to have a rigorous definition of the area under a curve that can be used to formalise the statement of the fundamental theorem of calculus.
It seems natural to proceed in the same spirit: teach Bayesian updating, and drill our students in these methods of calculation. The fact that there is only one updating rule that really works is mentioned but not proved. Those who follow the maths track get to see Cox’s theorem two or three years later.
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.
My view is that writing a textbook on Bayesian statistics is very difficult because it is hard to order the material in a satisfactory way.
Here is why I’m wrong: when we teach calculus we teach differentiation. Then we say that integrals are important because they are areas under curves. Then we drill our pupils in the computation of integrals by finding anti-derivatives. It is only two or three years later that we introduce the Riemann integral in order to have a rigorous definition of the area under a curve that can be used to formalise the statement of the fundamental theorem of calculus.
It seems natural to proceed in the same spirit: teach Bayesian updating, and drill our students in these methods of calculation. The fact that there is only one updating rule that really works is mentioned but not proved. Those who follow the maths track get to see Cox’s theorem two or three years later.
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.