6-ConceptsProbability Page 166 Wednesday, February 4, 2004 3:00 PM
166 The Mathematics of Financial Modeling and Investment Management■ Probability as “relative frequency” as formulated by Richard von Mises.^3
■ Probability as an axiomatic system as formulated by Andrei N. Kol-
mogorov.^4The idea of probability as intensity of belief was introduced by John
Maynard Keynes in his Treatise on Probability. In science as in our daily
lives, we have beliefs that we cannot strictly prove but to which we
attribute various degrees of likelihood. We judge not only the likelihood of
individual events but also the plausibility of explanations. If we espouse
probability as intensity of belief, probability theory is then a set of rules
for making consistent probability statements. The obvious difficulty here is
that one can judge only the consistency of probability reasoning, not its
truth. Bayesian probability theory (which we will discuss later in the chap-
ter) is based on the interpretation of probability as intensity of belief.
Probability as relative frequency is the standard interpretation of
probability in the physical sciences. Introduced by Richard Von Mises in
1928, probability as relative frequency was subsequently extended by
Hans Reichenbach.^5 Essentially, it equates probability statements with
statements about the frequency of events in large samples; an unlikely
event is an event that occurs only a small number of times. The difficulty
with this interpretation is that relative frequencies are themselves uncer-
tain. If we accept a probability interpretation of reality, there is no way
to leap to certainty. In practice, in the physical sciences we usually deal
with very large numbers—so large that nobody expects probabilities to
deviate from their relative frequency. Nevertheless, the conceptual diffi-
culty exists. As the present state of affairs might be a very unlikely one,
probability statements can never be proved empirically.
The two interpretations of probability—as intensity of belief and as
relative frequency—are therefore complementary. We make probability
statements such as statements of relative frequency that are, ultimately,
based on an a priori evaluation of probability insofar as we rule out, in
practice, highly unlikely events. This is evident in most procedures of
statistical estimation. A statistical estimate is a rule to choose the proba-
bility scheme in which one has the greatest faith. In performing statisti-
cal estimation, one chooses the probabilistic model that yields the(^3) Richard von Mises, Wahrscheinlichkeitsrechnung, Statistik unt Wahrheit (Vienna:
Verlag von Julius Spring, 1928). (English edition published in 1939, Probability, Sta -
tistics and Truth.)
(^4) Andrei N. Kolmogorov, Grundbegriffe der Wahrscheinlichkeitsrechnung (Berlin:
Springer, 1933). (English edition published in 1950, Foundations of the Theory of
Probability.)
(^5) At the time, both were German professors working in Constantinople.