De Finetti's Coherence Principle and Frequentist interpretation

Question

So, without proof or citation, I often see that the Coherence Principle by de Finetti does not hold with Frequentist statistics. It is pretty easy to create examples of this fact. The exception would be where the Frequentist and the Bayesian results map to the same values everywhere.

Can someone cite a proof for me, or provide one as the general case? I have looked and failed to find one in the literature.

EDIT Proof that if I were to gamble that $\mu\ge{5}$ and construct the odds using a Frequentist method, then there would exist cases where a Dutch Book could be constructed. That would except the case where the Bayesian and Frequentist solution map to the same answer.

As to how do you apply coherence to Frequentist statistics, you don't. However, econometrics does. Finance is almost purely a Frequentist discussion. Less than one percent of finance articles are Bayesian and they usually cover side cases. Models like Black-Scholes, aside from being problem-ridden, are also Frequentist in construction.

I can show that in my specific case, there will arise cases where a Dutch Book can be constructed. What I was hoping for was a general proof.

I had assumed the fact that the Dutch Book Theorem does not hold in finance was probably not a problem in real-world activity, but I am pretty certain that I was wrong.

Answer 1

Savage has a result about any procedure for making decisions under uncertainty that satisfied a set of reasonable-looking conditions including coherence. He showed it implied the existence of finitely-additive beliefs over events and linear utilities over states of nature. There's a description here

If you have well-defined priors over everything in the world I don't think it's controversial that you would update them using Bayes' Theorem, so this gets basically at the same point that non-Bayesian inference must fail to be coherent.

The 'Dutch Book' argument is related but slightly different: it says that if you identify degrees of belief in uncertain statements with willingness to take certain bets, and if the degrees of belief aren't Bayesian probabilities, someone can find a set of bets you will accept that has negative value with probability 1 and take all your money.

The problem with the Dutch Book argument is that it assumes everyone should always be willing to take one or other side of every offered bet. Since they clearly aren't, the claim that they rationally must be is a hard sell, and one can argue that the whole problem is an artifact of identifying degrees of belief with willingness to bet. That's combined with the lack of any real evidence that frequentists are subject to Dutch Book tricks to siphon away all their money.

If you regard the betting part of the argument as just a metaphor and not really about betting or money you are basically back at Savage's argument.

Savage's argument is, I think, regarded as more persuasive, at least in an ideal sense. The reason it doesn't convince people that Bayesian statistics is the only way in practice, is that it assumes you are already able to make coherent decisions over all possible events (in order to have coherent priors). In that case you wouldn't need Bayesian statistics, you'd naturally just update your beliefs properly. Perfectly rational beings would automatically be Bayesian, but it doesn't necessarily follow that the best strategy for imperfect humans is to try to do formal Bayesian inference. It's a plausible supporting argument, but the proof isn't as airtight as people sometimes argue.

Answer 2

Frequentist inference can be (intra-experimentally) coherent without having the same results as a Bayesian. For a modern example, see: https://arxiv.org/abs/2111.10715