John DiNardo 117“collective” is beyond our scope, it is important to acknowledge that there can
be “legitimate” disagreements about whether certain probabilities can be said to
“exist.” Von Mises argues that the reason it is possible to talk about the probability
of a tossed coin turning up “Heads” is because it is easy to think of the “collec-
tive”; it is not possible, he says, to consider “the probability of winning a battle
...[which] has no place in our theory of probability because we cannot think of a
collective to which it belongs.”
I personally share von Mises discomfort with defining the “probability of
winning a battle,” but I imagine others do not. Whether or not it would be “mean-
ingful” to do so, or whether it “has no place in our theory of probability,” the
ultimate criterion in the non-Bayesian context is: “Would doing so help in under-
standing?” The salient issue is not that different, in principle, from the qualms a
physicist might feel about “whether a particular mechanical system is in fact iso-
lated or not.” Whether that is a “defect” of the theory of probability or whether
it introduces an undisciplined element of “subjectivity” is a subject upon which
there has been much philosophical debate.^29
3.4.5 Epistemic probability
The difficulties that a non-Bayesian might feel about conceiving of the appropriate
collective are largely avoided/evaded when we consider a different notion of prob-
ability – epistemic. A nice place to start is a description from Savage, often called a
“radical subjectivist:”
You may be asking, “If a probability is not a relative frequency or a hypothetical
limiting relative frequency, what is it? If, when I evaluate the probability of
getting heads when flipping a certain coin as .5, I do not mean that if the coin
were flipped very often the relative frequency of heads to total flips would be
arbitrarily close to .5, then what do I mean?” We think you mean something
about yourself as well as about the coin. Would you not say, “Heads on the next
flip has probability 0.5” if and only if you would as soon guess heads as not,
even if there were some important reward for being right? If so, your sense of
“probability” is ours; even if you would not, you begin to see from this example
what we mean by “probability.” (Savage, 1972)What is also interesting is that instead of Kolomogorov’s axioms reflecting a
(possibly) arbitrary set of axioms about unknown concepts which (one hopes)
resemble some real world situation, they can also be derived from “betting rules.”
Again quoting Savage:
For you, now, the probabilityP(A)of an eventAis the price you would just be
willing to pay in exchange for a dollar to be paid to you in caseAis true. Thus,
rain tomorrow has probability 1/3 for you if you would pay just $0.33 now in
exchange for $1.00 payable to you in the event of rain tomorrow. (ibid.)