Probabilities: A Historical Perspective - Information Fusion in Signal and Image Processing - page 258

human reasoning and the models used to describe it gained interest and led researchers
to return to a more subjectivist conception of probabilities. Two schools of thought
have appeared, one for additive properties and the other for non-additive properties.
The first school of thought relies on basic postulates to more rigorously and less
arbitrarily obtain probabilities, their properties, Bayes' theorem, etc. Among them
we can mention the works of Keynes [KEY 29], Kemble [KEM 42], Cox [COX 46],
Jaynes [JAY 57], Jeffreys [JEF 61], Tribus [TRI 72]. We will discuss Cox's approach
in detail in section A.3. Tribus's approach is directly inspired by it. Jaynes and Kemble
worked in the field of statistical mechanics and showed that this requires a subjectivist
approach. Jeffreys, based on a method similar to Cox's but in which numbers are
included by convention, inferred the properties of probabilities from a certain number
of principles. These principles reject others considered to be fundamental in other the-
ories (for example, the definition of probabilities in terms of infinite sets of possible
observations, in terms of world properties, the causality principle, etc.). The essence of
his theory is that none of the direct probabilities, whether a priori or a posteriori ,isa
frequency. Even if the probability is calculated based on a frequency, it is not identical
to the frequency and a reasonable degree of confidence is necessary before it can be
used. The goal of Jeffreys's theory was not to justify inductive reasoning, but to ensure
its mathematical consistency. In a very similar fashion, a great number of researchers
studied the philosophical perspective of subjective probabilities with respect to objec-
tive probabilities, often without questioning additivity (Keynes, Jeffreys, Ramsey, de
Finetti, Koopman, Russel, Carnap, Good, Savage, etc.). In particular, the works of
Savage and Finetti showed that the Bayesian theory's subjective approach is the most
justified and the most consistent [FIN 37]. In his works, de Finetti adopted a resolutely
subjective approach (more than that of Cox) and reasoned in terms of the consistency
of individual opinions, and even in terms of collective psychology in order to explain
the coincidences in the opinions of different individuals. This method, which is par-
ticularly interesting, is described briefly in section A.3.
The second school of thought completely re-examined additivity, relying in par-
ticular on the works of pioneers such as Bernoulli and Lambert (see, for example,
[GOO 59, SHA 86]). Koopman, in the 1940s, introduced the concept of lower and
upper probabilities, thus defining a subjective probability with an inequality and no
longer as a precise value, based on the works of Boole ( Laws of Thought , 1854),
which had already foreseen this evolution. Several other researchers followed up on
his work (Good, Dempster, etc.). In particular, Dempster generalized Lambert's rules,
which are only able to deal with arguments involving a single conclusion, in the case
when several hypotheses need to be considered. Applications of these new theories
can be found in the field of economics, where Shackle, for example, suggested eco-
nomic models relying on concepts close to possibility theory, or in the field of legal
precedents, particularly the works of Ekelöf ( Rättegang , 1963) who suggested three
operators for combining testimonies: the combination of consecutive testimonies and