Image Processing Reference
In-Depth Information
contrary, accepted indeterminism very well. To sum up, all of these works dealt with
the problem of games and produced a theory of chance, but probability is never men-
tioned, even if the vocabulary used is slightly more epistemological. There was still
significant confusion between statistics and prior knowledge, which led to two classes
of probabilities being confused, based on frequencies (related to statistics) and based
on equiprobable cases (related to prior knowledge).
A.1.2. Towards the Bayesian mathematical formulation
The first link between game theory and probabilities was given in 1662, when it
was introduced by Arnauld in the Art of Thinking . Arnauld established an analogy
between games and everyday life and suggested that an epistemological perspective
of chance made it possible to apply the theory to probabilities (which were still consid-
ered as attributes of opinion). He stopped short of the concept of numerical probability,
but his works clearly mark a milestone in the evolution of the concept of probability.
The analogy between games and life was used up until the end of the 17 th century by
demographers who calculated life expectancy tables by using game theory, but without
including the concept of probability.
A contribution from an entirely different field came from Leibniz, who suggested
in the De Conditionibus (1665) to represent a person's legal rights using numbers.
The absence of law was represented by 0, a pure law by 1 and a conditional law by
a fraction between 0 and 1. This classification of rights relies on the condition upon
which the law is founded: an impossible condition leads to the absence of law, if it is
necessary the law is pure, if it is contingent 2 , the law is conditional. These concepts
of contingency and necessity are also used by Bernoulli regarding the problem of the
combination of testimonies. Leibniz suggests relating the probability for the condition
to exist with the “quantity” of the law and thus seemed to lean towards a numerical
conception of probability, without relying on game theory. He only later becomes
acquainted with this theory. Although his essays on chance provided nothing new
from a mathematical perspective, they acknowledged the link between probability and
game theory.
In the field of the combination of testimonies, the works of Hooper in 1699 ( A Cal-
culation of the Credibility of Human Testimony ) lead to the definition of non-Bayesian
confidence functions, which represent the credibility of a witness, as well as to two
combination rules, one for consecutive testimonies and the other for simultaneous
testimonies. These two rules, which were very popular in the 18 th
century, were com-
pletely abandoned in the 19 th
2. Contingent is used here meaning that something may or may not occur.
Search WWH ::

Custom Search