Evaluation of Possibility

Evaluation of Possibility

Given a finite set of reference $\Omega$ , we assign to each event defined over $\Omega$ , i.e. to any subset of $\Omega$ , a coefficient ranging between 0 and 1 which evaluates how possible this event is:

value 1 means that the event is completely possible,
0 means that the event is impossible.

To define this coefficient, we introduce a possibility measure noted $\Pi$ , which is a function defined over set $\mathcal{P}(\Omega)$ of the parts of $\Omega$ , taking its values in [0, 1], such that:

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

(31)

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

(32)

If only two parts are considered, property 32 becomes reduced to:

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

(33)

This expresses that the occurrence of one of the two events A or B, taken indifferently, receives the same coefficient of possibility as the occurrence of the most possible event.

An event is completely possible if the measurement of its possibility is equal to 1, and impossible if it is null.

It can be noted that the coefficient assigned to the intersection of events, i.e. the value of the possibility measure associated to the intersection of parts of $\Omega$ , is not given by conditions 31 and 32 which define this measure. These conditions impose however that the coefficient is always increased by the smaller of the coefficients assigned to each of the two events.

A measurement of possibility thus satisfies:

$\begin{displaymath}\forall (A, B) \in \mathcal{P}(\Omega)^2, \quad\Pi(A \cap B) \leqslant \min(\Pi(A), \Pi(B))\end{displaymath}$

(34)

In particular, two events can be possible ( $\Pi(A) \neq 0, \Pi(B) \neq 0$ ), but their simultaneous occurrence impossible ( $\Pi(A \cap B) = 0$ ).

It is also deduced that, if we study an unspecified event on $\Omega$ and its opposite, at least one of them is completely possible. This means that any part A of $\Omega$ satisfies $\Pi(A) = 1$ or that its complement $\complement_A$ relative to $\Omega$ itself satisfies $\Pi\left(\complement_A\right) = 1$

The following properties are satisfied:

$\displaystyle \forall A \in \mathcal{P}(\Omega),$

$\displaystyle \max\left(\Pi(A), \Pi\left(\complement_A\right)\right) = 1$

(35)

$\displaystyle \Pi(A) + \Pi\left(\complement_A\right) \geqslant 1$

(36)

It is noticed that the coefficient of possibility assigned to a part A of $\Omega$ influences moderately the coefficient assigned to its complement $\complement_A$ . If the first is not equal to 1, the second must be; otherwise it has an unspecified value.

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