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