Capacities Determination - Probabilistic Composition of Preferences, Theory and Applications - page 94

Chapter 9
Capacities Determination

Abstract A capacity may be derived from the observed evaluations by following
the principle of maximizing posterior probabilities. By this choice, of the posterior
probability as paradigm, will be assigned highest capacity to those sets of criteria
for which is highest the probability of some alternative maximizing the preference.
Thus more importance is assured to those criteria with highest power of isolated
discriminating a best option and if criteria repeat each other their importance is not
magni

ed by such repetition.

Keywords Capacity Choquet integral Estimation Preference by at least one

criterion IDH

9.1 The Maximization Capacity

A capacity for a set of criteria is here derived from the probabilities of maximizing
preference according to them. The finality of adopting such form of derivation is to

assign highest capacity to those sets of criteria for which is highest the probability
of some alternative maximizing the preference.
The rule to derive the capacity from the probabilities of preference maximization
will consist of

first computing, for each subset of the set of criteria, the probabilities
of maximization according to at least one of the criteria of the subset. The capacity
will be proportional to the maximum, along the alternatives, of these probabilities
of preference maximization. Its final value will be obtained by proportionally

rescaling this vector of maxima in such way to give the value 1 to the set of all the
criteria.
Formally, the capacity estimation algorithm to generate the capacity of a subset
C 1 ; ... ;

f of s criteria has the two following steps:

First compute

C s g

P ð f C 1 ; ... ; C s g Þ ¼ max

a

ð 1 ð 1 P a1 Þ ... ð 1 P as Þ Þ;