Possibility distribution

Let U a set of elementary events, $u$. One calls possibility measure denoted $\Pi$ a function defined on the set of all parts $P(U)$ of $U$, taking values in $[0, 1]$ such as:

A possibility degree $\Pi(A)=1$ means that the event $A$ is completely possible, inversely $\Pi(A)=0$ means that $A$ is impossible.

A possibility distribution assigns to each element $u$ of $U$ a possibility
$\pi(u) \in [0, 1]$. The distribution is normalized: $\sup_{u \in U}\pi(u) = 1$.

Guaranteed possibility

A guaranteed possibility measure $\Delta$ is a function defined on the set of all parts of $U$, $[0, 1]$ taking values in $[0, 1]$ , such as:

Relation between possibility degree and guaranteed possibility degree:

$\displaystyle \forall A \subseteq U, \Delta(A) = inf_{u \in A} \pi(u) $

.

Thus, if $\Delta(A) = \alpha$, all elementary events $u \in A$ are guaranteed possible at level $\alpha$.

Guaranteed possibility distributions are denoted $\delta$.

Interpretation of possibility degree and guaranteed possibility degree