Evaluation of Necessity

Evaluation of Necessity

A possibility measure provides information on the occurrence of an event A relating to a reference set $\Omega$ . It is not enough however to describe existing uncertainty on this event.

For example: let us assume that $\Pi(A) = 1$ , which means that it is completely possible that A occurs. But one can have at the same time $\pi\left(\complement_A\right) = 1$ . This expresses a complete indetermination on the occurrence of A. On the other hand, if $\pi\left(\complement_A\right) = 0$ , then only fact A can be fulfilled. Its occurrence is thus certain.

To complement information on A, we introduce the degree with which the occurrence of A is certain, by means of a necessity measure, a complement to the possibility measure.

This necessity measure also assigns a coefficient ranging between 0 and 1 to any part of $\Omega$ , but its properties are different from those of a possibility measure.

A necessity measure N is a function defined on set $\mathcal{P}(\Omega)$ of the parts of $\Omega$ , with values in [0, 1], such that:

$\begin{displaymath}N(\varnothing) = 0, \quad N(\Omega) = 1\end{displaymath}$

(37)

$\begin{displaymath}\forall A_1 \in \mathcal{P}(\Omega),\forall A_2 \in \mathc......= 1, 2, \ldots} A_i \right)= \inf_{i = 1, 2, \ldots} N(A_i)\end{displaymath}$

(38)

If only two parts are considered, property 38 is reduced to:

$\begin{displaymath}\forall (A,B) \in \mathcal{P}(\Omega)^2, \quadN(A \cap B) = \min(N(A), N(B))\end{displaymath}$

(39)

This definition imposes the monotony of N. Part A of $\Omega$ which contains part B has a necessity measure greater than B:

$\begin{displaymath}\text{if } B \subseteq A, \quad \text{then } N(B) \leqslant N(A)\end{displaymath}$

(40)

The value of the necessity measure associated with the union of parts of $\Omega$ is not given but, because of this monotony, it satisfies:

$\begin{displaymath}\forall (A,B) \in \mathcal{P}(\Omega)^2,N(A \cup B) \geqslant \max(N(A), N(B))\end{displaymath}$

(41)

A connection is deduced between the necessity measure of an unspecified event A and its complement $\complement_A$ :

$\begin{displaymath}\forall A \in \mathcal{P}(\Omega), \quad\left\{ \begin{arra...... \\N(A) + N(\complement_A) \leqslant 1\end{array} \right.\end{displaymath}$

(42)

This connection is relatively low since, if N(A) is not null, then $N(\complement_A)$ must be null, otherwise $N(\complement_A)$ can take any value.

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