Is the criterium of admissibility a valid requirement for probability interpretations?

Question

One argument that is often raised against hypothetical frequentism, as e.g. developed by Reichenbach or Von Mises is that limiting relative frequencies can violate the axiom of countable additivity (see e.g. Alan Hajek's 15 arguments against hypothetical frequentism) and are therefore not admissible. Where admissible is a requirement brought forward by Salmon, which demands that:

We say that an interpretation of a formal system is admissible if the meanings assigned to the primitive terms in the interpretation transform the formal axioms, and consequently all the theorems, into true statements. A fundamental requirement for probability concepts is to satisfy the mathematical relations specified by the calculus of probability...

I would question however that this is really a valid requirement. If the interpretation of probability as limiting relative frequency is helpful and well-working in many cases, why should these be spoiled by the fact that it fails in some, possibly irrelevant cases? The following example might clarify my idea: If certain assumptions are met, I can interpret linear regression as causal relationships between variables. For this to be a valid interpretation it is however irrelevant whether there are causal relationships that do not fit into the framework of linear regression. The same holds for hypothetical frequentism: Just because there are imaginable scenarios with arising limiting relative frequencies that do not fit into Kolmogorov's axiomatization it does not mean that I can validly interpret probability as limiting frequencies in many other cases. Do you think I have a valid point or am I missing an important argument for the admissibility criterion?

Answer 1

The argument you are referring to is Argument 13 of Hajek (2009) (pp. 229-230). It shows that it is possible to form a sequence of outcomes where the limiting relative frequency of each possible outcome is zero. Hajek argues that strikes against the frequentist interpretation of probability, because it means that the probabilities of all possible outcomes must be zero, which means that one cannot satisfy both the norming axiom and countable additivity axiom simultaneously. This argument is certainly correct, insofar as it goes, but a frequentist might object that the outcome that is used is "improbable" (in some as yet undefined sense) in a hypothetically repeated process. This is a challenge for the frequentist though, and it is potentially fatal.

Personally, I think that the best resolution to understanding the relationship between "probability" and "limiting relative frequency" comes from the Bayesian modelling approach, based on an IID model formed from an exchangeable sequence (see e.g., O'Neill 2009 for discussion). If you have an exchangeable sequence of values and you take a subjective interpretation of probability, then it can be shown that ---with probability one--- the limiting relative frequency of any outcome will correspond to the marginal probability of that outcome. (This is encapculated in the famous "representation theorem" of de Finetti, extended by others.) So one way to view this problem is to say that probability should have a "subjectivist" interpretation a-la Bayesian statistics, but in the case of an exchangeable sequence (which represents a repeatable trial where order is non-informative), this "probability" coincides with the "limiting relative frequency" (with probability one) and thus, the frequentist interpretation is also correct in this context (with probability one).

This latter interpretation resolves Hajek's objection because under the condition of exchangeability, the objectionable series he is referring to has (subjective) probability zero. Thus, in the above model, we know that the frequentist equivalence holds with (subjective) probability one, and the objectionable examples (e.g., Hajek's Argument 13 example) have total (subjective) probability zero.