116 Metastatistics for the Non-Bayesian Regression Runner
prototype for what we have in mind:
HTHTHT...HT...In such an example, if we know the last coin toss was “H” we are certain that the
next coin toss will be “T.” An “intuitive” definition that excludes such a possibility
was given by Venn (1888), who talked about a probability as a characteristic of a
seriesas “one which exhibits individual irregularity along with aggregate regular-
ity.” If we denote the number of “Trials” byNand the number of times the event
“Heads” occurs asm(N), we might go on to define the probability of “Heads” as:
P(Head)= lim
N→∞m(N)
N
.An apparent weakness of this definition is, of course, that infinity is rarely
observed. The derisive term about such thought exercises is sometimes referred to
as “asymptopia” – which suggests something both unrealistic and unattainable.^27
3.4.4 Objective, subjective, or “it depends”
Whether such a concept corresponds to something “real” or “objective,” or
whether it is “in the mind,” is a subject on which much has been written. Some-
times such probabilities have been described as “objective” in order to contrast
them with Bayesian “probabilities.” However, one Bayesian objection is that there
is no such thing as an “objective probability” – any such probability depends on
purely subjective beliefs:
To calculate a frequency, it is necessary to consider a repetitive phenomenon
of a standardized variety. The appropriate meaning of “repetitive” and “stan-
dardized” is not obvious. To calculate a relative frequency it is necessary to
subjectively define (possibly only conceptually) a class of events (known as a
collective) over which to count the frequency. The relative frequency of the event
[“Heads”] should also tend to the same limit for all subsequences that can be
picked out in advance.^28
To the extent that individuals agree on a class of events, they share an
objective frequency. The objectivity, however, is in themselves, not in nature.
(Poirier, 1995)Poirier, a Bayesian, stresses the (implicit) “subjectivity” of the frequentist notion of
probability, specifically the notion of a “collective.” The non-Bayesian von Mises
(1957, p. 12), for example, defines a collective as a “sequence of uniform events
or processes which differs by certain observable attributes, say colours, numbers,
or anything else. Only when such a collective is defined, then a probability can
be defined. If it is impossible to conceive of such a collective, then it is impossible
to talk about probability.” For von Mises, the notion of collectives with infinite
numbers of entities was anabstractionto make the mathematical representation
of reality “tractable” (Gillies, 2000, p. 90). While an extensive discussion of a