6-ConceptsProbability Page 167 Wednesday, February 4, 2004 3:00 PM
Concepts of Probability 167highest probability on the observed sample. This is strictly evident in
maximum likelihood estimates but it is implicit in every statistical esti-
mate. Bayesian statistics allow one to complement such estimates with
additional a priori probabilistic judgment.
The axiomatic theory of probability avoids the above problems by
interpreting probability as an abstract mathematical quantity. Devel-
oped primarily by the Russian mathematician Andrei Kolmogorov, the
axiomatic theory of probability eliminated the logical ambiguities that
had plagued probabilistic reasoning prior to his work. The application
of the axiomatic theory is, however, a matter of interpretation.
In economics and finance theory, probability might have two differ-
ent meanings: (1) as a descriptive concept and (2) as a determinant of
the agent decision-making process. As a descriptive concept, probability
is used in the sense of relative frequency, similar to its use in the physical
sciences: the probability of an event is assumed to be approximately
equal to the relative frequency of its occurrence in a large number of
experiments. There is one difficulty with this interpretation, which is
peculiar to economics: empirical data (i.e., financial and economic time
series) have only one realization. Every estimate is made on a single
time-evolving series. If stationarity (or a well-defined time process) is
not assumed, performing statistical estimation is impossible.PROBABILITY IN A NUTSHELL
In making probability statements we must distinguish between outcomes
and events. Outcomes are the possible results of an experiment or an obser-
vation, such as the price of a security at a given moment. However, proba-
bility statements are not made on outcomes but on events, which are sets of
possible outcomes. Consider, for example, the probability that the price of
a security be in a given range, say from $10 to $12, in a given period.
In a discrete probability model (i.e., a model based on a finite or at
most a countable number of individual events), the distinction between
outcomes and events is not essential as the probability of an event is the
sum of the probabilities of its outcomes. If, as happens in practice,
prices can vary by only one-hundredth of a dollar, there are only a
countable number of possible prices and the probability of each event
will be the sum of the individual probabilities of each admissible price.
However, the distinction between outcomes and events is essential
when dealing with continuous probability models. In a continuous proba-
bility model, the probability of each individual outcome is zero though the
probability of an event might be a finite number. For example, if we repre-