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2.3.3 Possibility theory and fuzzy measures
In the late 1970s Zadeh advanced a theoretical framework for information and knowledge
analysis, called possibility theory, emphasising the quantification of the semantic,
context-dependent nature of symbols—that is, meaning rather than measure of
information (Tsoukalas and Uhrig, 1997). The theory of possibility is analogous to, yet
conceptually different from the theory of probability. Probability is fundamentally a
measure of the frequency of occurrence of an event, while possibility is used to quantify
the meaning of an event.
To understand the distinction between the probability and possibility of an event, it is
worthwhile to observe an example similar to that given by Zadeh (1978). Suppose that
we have a proposition “Hans ate X eggs for breakfast”, where X ={1, 2, 3…}. The
probability distribution, p X (x), and the possibility distribution, r X (x), associated with X
may be represented as given in Table 1.
Table 1: An example of probability and possibility distributions of an event.
No of eggs (x)
1
2
3
4
5
6
7
8
9
p X (x)
0.2
0.7
0.1
0
0
0
0
0
0
r X (x)
1
1
0.8
0.6
0.3
0.2
0.1
0.0
0.0
The possibility distribution is interpreted as the degree of ease with which Hans eats x
eggs. For example, the degree of ease with which Hans can eat two eggs is 1, whereas the
degree of ease for eating seven eggs is only 0.1. The probability distribution might have
been determined by observing Hans at breakfast for several days. As can be seen in Table
1, the probabilities in a probability distribution must sum to unity. The possibility
distribution, on the other hand, is the situation or context-dependent and does not have to
sum to unity. It is observed from the table that a high degree of possibility does not imply
a high degree of probability. If, however, an event is not possible, it is also improbable. Thus,
in a way, possibility is an upper bound for the probability. This weak connection between
the two is known as possibility/probability consistency principle after Zadeh (1978).
Zadeh (1978) further postulated possibility as a fuzzy measure . A fuzzy measure is a
function with a value between 0 and 1, indicating the degree of evidence or belief that a
certain element x belongs to a set (Zadeh, 1978; Tsoukalas and Uhrig, 1997). Therefore
the membership function of a fuzzy event, say A, can be considered as a possibility
distribution of the event. That is,
r A (x) = " A (x)
(2.17)
It should be noted however that only the normal membership function defines possibility
measures (Bardossy and Duckstein, 1995). By virture of this connection between
possibility distribution and the fuzzy membership function, the treatment of uncertainty
using the fuzzy set theory is broadly known as a possibilistic approach (Langley, 2000).
 
 
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