118 Metastatistics for the Non-Bayesian Regression Runner
As when we encountered Expected Utility, viewing probability as a device that
allows one to make sensible “bets” is notnecessary.The important distinction
between aleatory and epistemic probability is that epistemic probabilities are num-
bers which obey something like Kolmogorov’s axioms but do not refer to anything
“real” in the world, but to a (possibly) subjective “degree of belief.” Here’s one
definition from Poirier (1995, p. 19):
Let κ denote the body of knowledge, experience or information that
an individualhas accumulated about the situation of concern, and letAdenote
an uncertain event (not necessarily repetitive). Then theprobabilityafforded by
κis the “degree of belief” inAheldby the individualin the face ofκ.Given this definition of probability, stating that the probability that a fair coin
lands heads isnotstating some property of a chance set-up – rather, it is an
expression of belief about what the coin will do.^30 It is important to point out
that opinions about this subject vary amongst Bayesians. I.G. Good, for instance,
maintains that “true” probabilities exist but that we can only learn about them by
using subjective probabilities. DeFinetti, as we saw, believes that it is unhelpful to
postulate the existence of “true” probabilities.
How does this differ from the aleatory- or frequency-type probability we
discussed above? Again quoting from Poirier (1995, p. 19):
According to the subjective...interpretation, probability is a property of
an individual’s perception of reality, whereas according to the...frequency
interpretations, probability is a property of reality itself.Among other things, in this view the probability that a fair coin-toss is heads
differs across individuals.^31
3.4.6 Conditional probability, Bayes’ rule, theorem, law?
Is it Bayes’ rule, law, or theorem? Is it one of the most powerful ideas of all time,
or the source of much mischief? Dennis Lindley (as cited in Simon, 1997) observes
that “[Bayes’] theorem must stand with Einstein’sE=mc^2 as one of the great, sim-
ple truths.” Putting aside the intractable issue of what the Reverend Bayesmeant,
this has been the subject of considerable controversy and study.^32
For the typical non-Bayesian, R.A. Fisher and William Feller, for example, Bayes’
rule is nothing but a manipulation of the law of conditional probability.
Everyone starts with adefinitionof conditional probability:
P(Ai|B)=
P(Ai⋂
B)
P(B)
ifP(B)>0. (3.2)Provided the necessary probabilities exist, we can do the same thing in reverse:
P(B|Ai)=P(Ai
⋂
B)
P(Ai)
ifP(Ai)>0. (3.3)