John DiNardo 115This too could form the basis of a theory of probability, but it is one we choose not
to adopt because it seems “inconvenient” to work with and makes it more difficult
to study the behavior of chance set-ups.
3.4.2 Definitions of probability
DeFinetti’s declaration in Surprising Idea 6 that “PROBABILITY DOES NOT EXIST” may,
at minimum, appear to be a bit intemperate. Indeed, it presupposes that many
practically minded non-Bayesian regression runners are in the grips of some bizarre
hallucination. It will help to consider two broad classes of definitions of probability
that are sometimes referred to as:
- “aleatory” or frequency-type probabilities
- “epistemic” or belief-type probabilities.
Aleatory probabilities are perhaps what is most familiar to the non-Bayesian. For
many, the notion of any other type of probability may not have been seriously
entertained. It is interesting to observe that criticism of aleatory probability began
at the inception of modern statistics and, as Hacking (1975, p. 15) observes,
“philosophers seem singularly unable to put asunder the aleatory and the episte-
mological side of probability. This suggests that we are in the grip of darker powers
than are admitted into the positivist ontology.”
I began my presentation with Kolomogorov’s axioms since everyone seems to
agree on something like these; disputants disagree on what they are useful for,
or what, precisely, they are “about.” I won’t do a complete survey, but a few
moments of reflection may be all that is required to consider how slippery a notion
probability could be.^26
3.4.3 Aleatory or frequency-type probabilities
When we say “the probability that a fair coin will land as heads is^12 ” we could take
it as a statement of fact, which is either true or not. When we do so, we are generally
thinking about probability as describing something that results from a mechanism
that tosses coins and the geometry of the coin, perhaps. This mechanism can be
described as a “chance set-up.” We might go on to describe the physics of the
place containing our coin-toss mechanism. A mechanism that would be perfectly
useful in Ann Arbor, Michigan, might not work somewhere in the deep reaches of
outer-space.
Nonetheless, most non-Bayesians, it would seem, are content to harbor little
doubt that, at some fundamental level – whether we know the truth or not – it
is meaningful to talk about the probability of a tossed coin falling heads. When
pressed to explain what they mean when they say that the probability is^12 that a
fair coin will turn up heads, such a person might say “In the long run, if I were to
repeatedly toss the coin in the same way, the relative frequency of heads would be
1
2 .” We’ve yet to worry about “the long run” but, even at this level, for example,
we would like to exclude the following deterministic but infinite series as being a