Probabilities: A Historical Perspective - Information Fusion in Signal and Image Processing - page 255

One of the most important contributions to the relation between game theory and
probability at the end of the 17 th century is probably that of Bernoulli, particularly in
his topic Ars Conjectandi (published in 1713). Bernoulli suggested as early as 1680
a mathematical theory of probabilities and their combinations. He uses game theory
to calculate probabilities, but does not remove from probabilities their epistemolog-
ical aspect and their role in the judgment of individuals. Thus, the first formalisms
that allowed numerical probabilities to be used focused on the study of subjective
probabilities! For the most part, Bernoulli's theory focuses on probabilities that are
subjective, and that are means of measuring knowledge. They are calculated based on
the concept of “arguments”, and their properties depend on the nature of these argu-
ments. In particular, they are not always additive. The combination rules suggested by
Bernoulli, which are more complete than Hopper's, take on different forms, depend-
ing on the arguments, and not all of these forms correspond to the usual probabilistic
rules. In the last part of his work, Bernoulli established his famous law of large num-
bers. This law allows a prior unknown probability to be estimated afterwards based
on the observation of occurrence frequencies. Therefore, this theorem has more to do
with random probabilities than epistemological probabilities (according to the distinc-
tion later introduced by Lambert), since a chance that can only be known a posteriori
is not initially a characteristic of our knowledge.
The successors of Bernoulli simplified his theory and unknowingly reduced its
scope. First of all, they were not convinced by Bernoulli whose theory, which often
seemed complicated to them, was not as well established as game theory. Addition-
ally, they mostly held on to the law of large numbers and identified probability with
the chance of appearing, thus leading to an essentially frequentist approach. We can
mention, among the successors of Bernoulli, Montmort ( Essai d'analyse sur les jeux
de hasard , 1708), who attempts to apply game theory to other fields, and Moivre ( De
Mensura Sortis , 1711, Doctrine of Chances , 1718) where we find the first explicit
additivity rule and a representation of probabilities between 0 and 1. His definition
became the classical definition. These two authors mentioned probabilities but their
theory often deals with chance. They defined probabilities as the ratio of the number
of favorable cases to the number of possible cases, but the authors were faced with the
problem of counting the number of cases, which is not possible in every field. Note
that Moivre had already discovered the Gaussian distribution.
In the 18 th century, only Lambert pursued Bernoulli's work and distinguished ran-
dom probabilities and epistemological probabilities. The former are those that can be
known a priori , as in game theory, or a posteriori , provided by experience. The latter
are assigned to events by inferences based on effects or circumstances and are more
subjective in nature. In Photometrica (1760), he focused on error theory and suggested
a method known today as maximum likelihood. In Neues Organon (1764), he general-
ized Bernoulli's argument theory, corrected and generalized his combination laws and
discussed the cases of games, syllogisms and of different types of testimonies. His