Probabilities in the Problem of Classification - Probabilistic Composition of Preferences, Theory and Applications - page 75

Chapter 8
Probabilities in the Problem
of Classication

Abstract Evaluation of alternatives may be derived from probabilistic comparisons
made with different sets of pro

les which represent ordered classes. This may be
used to solve the problem of, instead of choosing the best or worst alternative,
identifying alternatives that correspond to each of a set of different levels of

performance.

Keywords Classi cation Interval classi cation Pro le PseudoF IDH

8.1 Modeling the Problem of Classification

It was demonstrated in the preceding chapters that, while the computation of the
probabilistic scores of preference given by probabilities of being the best is easier if
the comparisons are made to a sample of

fixed previously chosen representative
alternatives instead of to the entire population of alternatives, the efficiency of the
evaluation will increase with the number of comparisons made. For these reasons, it
is important to balance the goal of reducing the number of comparisons to simplify
computations with that of increasing it to improve reliability.
In the problem of classi cation, addressed in this chapter, a large number of

comparisons is made, but the decisions are taken successively based on compari-
sons to small sets of possible alternatives previously ordered.
As in the problem of choice of the best or the worst, in the problem of classi-

fication the probabilistic approach is based on treating the initial numerical eval-
uations as location parameters of probability distributions.
To formulate the problem of classi

cation the following terms are employed.

G={g 1 , … ,g m }, a set of m criteria.

￿

,a m ) a vector of R m which stores the evaluations according to the m
criteria of the alternative to be classi

A=(a 1 , …

￿

ed; the highest the value of the coordinate
a j , the better the alternative according to the criterion g j .

￿ C={C 1 , ,C r } a set of r classes, ordered from the worst to the best, so that the

alternative is better if classi

…

ed in a class of higher index.