Image Processing Reference
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with intuitive postulates, which are directly related to what is expected of an induc-
tive logic 3 , from where they infer probability rules. This method was devised by Cox
[COX 46], essentially, and was discussed in detail, for example, in [DEM 93, PAR 95,
TRI 72], where demonstrations of the major results can also be found. Here, we will
present these fundamental postulates and the outline of the reasoning. At the end of
this section, we will present the works led by de Finetti [FIN 37]. Less known to signal
and images processors, they are appealing in two ways, both for their fundamentally
subjectivist aspect and for the simplicity of the demonstration.
A.3.1. Fundamental postulates
Here are the fundamental postulates as laid out by Cox [TRI 72] (those suggested
by Jeffreys [JEF 61] are very similar 4 ):
1) consistency or non-contradiction : if a conclusion can be drawn in different
ways, they must all lead to the same result; there have to be no contradictory con-
clusions based on the same data; furthermore, equal confidences have to be attributed
to propositions that have the same truth value;
2) continuity of the method : the operations performed have to be continuous and if
a slight change occurs in the data, it must not lead to sudden changes in the result;
3) universality or completeness : it has to be possible to attribute a degree of confi-
dence to any well-defined propositions and to compare degrees of confidence;
4) unequivocal statements : propositions have to be well defined, i.e. it has to be
theoretically possible to determine whether a proposition is true or false. This is equiv-
alent to what Horvitz refers to as clarity [HOR 86];
5) no information is refused : conclusions cannot be drawn based on partial infor-
mation, meaning that all the information, experience or knowledge available related
to the proposition we wish to evaluate has to be taken into account and, in particular,
it is important to take into account the dependence of the context. This postulate is
3. The objective of inductive logic is to determine the most likely solution given the information
available, true and false being the extreme cases, as opposed to deductive logic, for which the
only possible cases are true, false, and total lack of knowledge.
4. Jeffreys, by trying to define a general inference method, laid out the following postulates: all
of the hypotheses have to be expressed, and the conclusions are inferred from the hypotheses;
the theory has to be consistent and not contradictory; every rule has to be applicable in practice;
the theory has to provide indicators for pointing out possible false inferences; the theory should
not systematically reject empirical information. Additionally, Jeffreys suggests relying on the
following guidelines: the number of postulates has to be kept to a minimum; the theory has to
be in agreement with human reasoning; because induction is more complex than deduction, we
cannot hope to develop it further than deduction. Jeffreys's approach then consists of translating
these postulates in more formal axioms, of introducing numbers to represent probabilities and
finally of demonstrating the traditional results [JEF 61].
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