Download Fundamentals of Uncertainty Calculi with Applications to by Michel Grabisch PDF

By Michel Grabisch

With the imaginative and prescient that machines might be rendered smarter, we have now witnessed for greater than a decade large engineering efforts to enforce clever sys­ tems. those makes an attempt contain emulating human reasoning, and researchers have attempted to version such reasoning from a variety of issues of view. yet we all know worthwhile little approximately human reasoning techniques, studying mechanisms and so forth, and particularly approximately reasoning with constrained, vague wisdom. In a feeling, clever platforms are machines which use the main normal type of human wisdom including human reasoning potential to arrive judgements. hence the overall challenge of reasoning with wisdom is the middle of layout method. The try to use human wisdom in its so much traditional feel, that's, via linguistic descriptions, is novel and debatable. the newness lies within the reputation of a brand new kind of un­ sure bet, particularly fuzziness in average language, and the controversality lies within the mathematical modeling method. As R. Bellman [7] as soon as stated, determination making less than uncertainty is without doubt one of the attributes of human intelligence. while uncertainty is known because the impossi­ bility to foretell occurrences of occasions, the context is standard to statisticians. As such, efforts to take advantage of likelihood conception as a necessary device for construction clever structures were pursued (Pearl [203], Neapolitan [182)). The method turns out all right if the doubtful wisdom in a given challenge could be modeled as likelihood measures.

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Extra resources for Fundamentals of Uncertainty Calculi with Applications to Fuzzy Inference

Example text

If in K, K n ~ K E K, then T*(Kn ) ~ T*(K), 4. if in C{Jn above, the K's are replaced by the A's E 2R , and T is replaced by T* , then for all n:::: 1, the new C{Jn are non-negative. To show that F(B) can be defined for all B E f3, the Borel a-field of IR, one can use the concept of T* -capacitable, which is similar to the concept of regular Radon measure on (IR, f3) (Meyer, [162]). Also, for [0, B] E A for all B E f3, one needs to replace (F, A, Q) by its completion. For ease of reference, we reproduce here the Choquet and Matheron theorems.

In connection with the example above, let JE :1. Then P(X n J -1= 0) = max{f(8) : 8 E J}. In particular, for 8 E 8, f(8) = Q{w : 8 E X(w)}. Thus f is the one-point coverage function (Goodman [78]) of the random set X. Following Goodman, one can associate with f a random set Y as follows. Let u be a random variable, uniformly distributed on [0,1]. Let Y(w) = {8 : u(w) ::::: f(8)}. Then for all 8 E 8, P(w : 8 E Y(w)) = P(w : u(w) ::::: f(8)) = f(B). If for tE [0,1], we let At = {B : f(B) :2: t} and C = {At: t E [0, I]}, we can consider the a-field a(C) on C given by B E a(C) if and only if for some Borel set A <;;; [0,1], B = {At : t E A}.

In fact, for an increasing set function I, strong subadditivity is equivalent to Ll2 :::; 0, where I replaces P in the definition of Ll above. Indeed, if Ll2 :::; 0, then Ll2(X; A, B) = I(X) - I(X U A) - I(X U B) + (X U A U B) :::; 0, and taking X = AB, we get I(A U B) + I(AB) - I(A) - I(B) ::; O. Conversely, substituting X U A and X U B for A and B, respectively, in the last inequality yields I(X U A U B) + I((X U A)(X U B)) - I(X U A) - I(X U B)) :::; 0, and since X ~ (X U A)(X U B) and I is increasing, we have that Ll2 :::; O.

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