Probability Applications

Timothy J. Ross, Kimberly F. Sellers, and Jane M. Booker 99
PowerSet(X) = {0, a, b, c, (a U b), (a U c), (b U c), (a U b U c)}. Figure 5.3 illustrates this
idea. In the figure, the universe of discourse is comprised of a collection of sets and subsets,
or the power set. In a fuzzy measure, what we are trying to describe is the ambiguity in
assigning this point x to any of the crisp sets on the power set. This is not a random notion;
it is a case of ambiguity. The crisp subsets that make up the power set have no uncertainty
about their boundaries as fuzzy sets do. The uncertainty in the case of a fuzzy measure
lies in the ambiguity in making the assignment. This uncertainty is usually associated with
evidence to establish an assignment. The evidence can be completely lacking — the case
of total ignorance — or can be complete and specific —the case of a probability assignment.
Hence the difference between a fuzzy measure and a fuzzy set on a universe of elements is
that in the former the ambiguity is in the assignment of an element to one or more crisp sets,
and in the latter the vagueness is in the prescription of the boundaries of a set. We see a
distinct difference between the two kinds of uncertainty, ambiguity and vagueness. Hence
what follows is a mathematical description of a theory very useful in addressing ambiguity —
possibility theory. This section is given for completeness in describing mathematical models
useful in addressing various forms of uncertainty and not as a matter of extending the scope
of this book beyond a comparison of the utilities of fuzzy set theory and probability theory.
Figure 5.3. The power set of a universe X.
To continue, a set function v(A) e [0, 1] for A c X is a fuzzy measure if A c B ->
v(A) < v(B). A triangular norm is a conjunctive operator n : [0, I] 2 —»• [0, 1], which is
associative, commutative, and monotonic, with identity 1. Typical examples are the multiplication
(jc * y) and minimum (min(;t, y)) operators. Their dual conorms are disjunctive, have
identity 0, and include bounded sum min(;c + _y, 1) and maximum max(;t, y). Norms and
conorms are used to operate on fuzzy measures in various ways. They also represent the different
degrees of dependence or independence among events in a probability space and thus
all possible relations among marginal and joint probability distributions (Schweizer (1991)).
A probability measure P is a fuzzy measure, where P(A U B) = P(A) + P(B) —
P(A n B}. The other major class of fuzzy measures consists of possibility measures ]~],
where f](A U B) = max(f](^), [~I(^))- Where a probability measure is characterized
by its probability distribution p(x) = P({x}) for jc e X such that P(A) = ^X&A p(x),
a possibility measure is characterized by its possibility distribution n(x) = Y[((x}) sucn
that H(^) — rnax x€/4 TT(JC). Also, while both probability and possibility measures are
generally normal, with P(X) — Y\(X) — 1, for a probability measure, this entails additivity