Probabilities: A Historical Perspective - Information Fusion in Signal and Image Processing - page 259

that of simultaneous corroborating testimonies follow Hooper's rules, whereas the
combination of conflicting testimonies is related to a specific case of Lambert's rule.
Starting in the 1960s, theories appeared that were no longer directly related to
probabilities. Zadeh invented fuzzy sets in 1965 [ZAD 65], Shortliffe and Buchanan
constructed the MYCIN system based on the concept of certainty factors in 1975
[SHO 75], Shafer developed belief theory ( A Mathematical Theory of Evidence , 1976)
[SHA 76] and Zadeh introduced possibility theory in 1978 [ZAD 78].
A.2. Objectivist and subjectivist probability classes
Section A.1 showed that there are several classes of probabilities, summed up here,
based on the classification set out by Good [GOO 59]:
1) the traditional definition is provided by game theory and relies essentially on
the concept of equally probable cases. Calculating the probabilities is achieved using
frequencies of occurrence, by counting all of the cases;
2) a more subjective version of this definition includes additional information
related to the knowledge we have, for example, knowledge of the honesty, or lack
thereof, of a game. Therefore, this second class only considers conditional probabil-
ities. We can also include in this class Savage's or Finetti's subjective probabilities,
which are estimated proportionally to the sum of money that a person would be will-
ing to give if what it claimed turned out to be false [DUB 88, FIN 37];
3) a third class is that of inverse probabilities, according to Bayes and Laplace.
It consists of the final probability of a hypothesis (after experiments have been con-
ducted) estimated from the a priori probability (in the absence of experiments) and the
conditional probability (or likelihood probability), or experiment probability given the
hypothesis;
4) the physical probabilities used in the 19 th century are no longer subjective in
nature and attempt on the contrary to achieve objectivity by calculating probabilities
that are conditional to experiments that have been conducted;
5) the purely frequentist approach calculates occurrence frequencies in large
sequences (Venn's limit, Fisher's infinite population, etc.);
6) finally, the last class, which Good calls “neo-classic” subjective probability, is
the largest. Probabilities represent degrees of confidence, with respect to a state of
knowledge, taking into account both subjective and objective information. This def-
inition encompasses all of the others and may be much more general. It relies on a
mathematical theory based on a few axioms, which makes it possible to ensure the
consistency of all the degrees of confidence. It can be extended to a theory of rational
behavior by including utilities. Finally, Good even suggests representing these sub-
jective probabilities with inequalities, which is similar to the Dempster-Shafer belief
theory, for example.