Composition by Joint Probabilities - Probabilistic Composition of Preferences, Theory and Applications - page 41

In case of a lack of strong reasons to consider correlation, the assumption of
independence should be preferred. This happens because, while the hypothesis of
independence leads to calculating the probability of the intersection using the
product of the probabilities of the events that are intersected, the hypothesis of
maximal dependence, because the probability of the intersection cannot be larger
than the probability of the events intersected, leads to calculating this probability
using the minimum of the probabilities of such events. Thus, the assumption of
independence, by involving the computation of the product of the exact values of
the factors, employs more of the available information.
It is interesting to notice here that the composition by the minimum is equivalent
to the application of the principle of necessity in Fuzzy Logic (Zadeh 1978 ).
Moreover, the hypothesis of maximal dependence provides a basis to establish a
relation between the composition by joint probabilities and that by integration, as
the combination by the minimum is equivalent to the application of a Choquet
( 1953 ) integral with respect to the capacity

(A) = 1 for every unitary set A.
When the number of criteria is large, the joint probability under the assumption
of independence will present small values as the product of a large number of
factors between zero and one. Under the assumption of maximal dependence,
determining the joint probability using the smallest of the probabilities according to
each criterion, the result will not be so small. However, simple procedures may be
employed to avoid small

μ with μ

final scores. It is enough to substitute for the product the
geometric mean, or to standardize the final probabilities to sum one, dividing each
product by the sum of all of them.
To illuminate the effect of the assumptions on dependence, the following for-
mulae show the result of application to the different points of view of the extreme
hypotheses of maximal dependence and independence. Here, M j and m j denote,
respectively, the probability of the alternative maximizing and minimizing the
preference according to the j-th criterion, π denotes the product, min denotes the

minimum and max denotes the maximum.

5.2.1 Progressive and Pessimistic Point of View

independence

: p j M j ;

maximal dependence : min j M j :

5.2.2 Progressive and Optimistic Point of View

;

independence 1 p j 1 M j

:

maximal dependence : max j M j :