Why Sugeno lambda-Measures

To describe expert uncertainty, it is often useful to go beyond additive probability measures and use non-additive (fuzzy) measures. One of the most widely and successfully used class of such measures is the class of Sugeno λ-measures. Their success is somewhat paradoxical, since from the purely mathematical viewpoint, these measures are – in some reasonable sense – equivalent to probability measures. In this paper, we explain this success by showing that while mathematically, it is possible to reduce Sugeno measures to probability measures, from the computational viewpoint, using Sugeno measures is much more efficient. We also show that among all fuzzy measures equivalent to probability measures, Sugeno measures (and a slightly more general family of measures) are the only ones with this property. I. FORMULATION OF THE PROBLEM Traditional approach: probability measures. Traditionally, uncertainty has been described by probabilities. In mathematical terms, probabilistic information about events from some set X of possible events is usually described in terms of a probability measure, i.e., a function p(A) that maps some sets A ⊆ X into real numbers from the interval [0, 1]. The probability p(A) of a set A is usually interpreted as the frequency with which events from the set A occur in real life. In this interpretation, if we have two disjoint sets A and B with A ∩B = ∅, then the frequency p(A ∪B) with which the events from A or B happen is equal to the sum of the frequencies p(A) and p(B) corresponding to each of these sets. This property of probabilities measures is known as additivity: if A ∩B = ∅, then p(A ∪B) = p(A) + p(B). (1.1) Need to do beyond probability measures. Since the appearance of fuzzy sets (see, e.g., [7], [9], [13]), it has become clear that to adequately describe expert knowledge, we often need to go beyond probabilities. In general, instead of probabilities, we have the expert’s degree of confidence g(A) that an event from the set A will actually occur. Clearly, something should occur, so g(∅) = 0 and g(X) = 1. Also, clearly, the larger the set, the more confident we are that an event from this set will occur, i.e., A ⊆ B implies g(A) ≤ g(B). Functions g(A) that satisfy these properties are known as fuzzy measures. Sugeno λ-measures. M. Sugeno, one of the pioneers of fuzzy measures, introduced a specific class of fuzzy measures which are now known as Sugeno λ-measures [10]. Measures from this class are close to the probability measures in the following sense: similarly to the case of probability measures, if we know g(A) and g(B) for two disjoint sets, we can still reconstruct the degree g(A∪B). The difference is that this reconstructed value is no longer the sum g(A) + g(B), but a slightly more complex expression. To be more precise, Sugeno λ-measures satisfy the following property: if A ∩B = ∅, then g(A ∪B) = g(A) + g(B) + λ · g(A) · g(B), (1.2) where λ > −1 is a real-valued parameter. When λ = 0, the formula (1.2) corresponding to the Sugeno measure transforms into the additivity formula (1.1) corresponding to the probability measure. From this viewpoint, the value λ describes how close the given Sugeno measure is to a probability measure: the smaller |λ|, the closer these measures are. Sugeno λ-measures have been very successful. Sugeno measures are among the most widely used and most successful fuzzy measures; see, e.g., [2], [11], [12] and references therein. Problem. This success is somewhat paradoxical. Indeed: • The main point of using fuzzy measures is to go beyond probability measures. • On the other hand, Sugeno λ-measures are, in some reasonable sense, equivalent to probability measures (see Section 2). How can we explain this? What we do in this paper. In this paper, we explain the seeming paradox of Sugeno λ-measures as follows: • Yes, from the purely mathematical viewpoint, Sugeno measures are indeed equivalent to probability measures. • However, from the computational viewpoint, processing Sugeno measure directly is much more computationally efficient than using a reduction to a probability measure. We also analyze which other probability-equivalent fuzzy measures have this property: it turns out that this property holds only for Sugeno measures themselves and for a slightly more general class of fuzzy measures. The structure of this paper is straightforward: in Section 2, we describe in what sense Sugeno measure is mathematically equivalent to a probability measure, in Section 3, we explain why processing Sugeno measures is more computationally efficient than using a reduction to probabilities, and in Section 4, we analyze what other fuzzy measures have this property. II. SUGENO λ-MEASURE IS MATHEMATICALLY EQUIVALENT TO A PROBABILITY MEASURE What we mean by equivalence. According to the formula (1.2), if we know the values a = g(A) and b = g(B) for disjoint sets A and B, then we can compute the value c = g(A ∪B) as c = a+ b+ λ · a · b. (2.1) We would like to find a 1-1 function f(x) for which p(A) def = f−1(g(A)) is a probability measure, i.e., for which, if c is obtained by the relation (2.1), then for the values a′ = f−1(a), b′ = f−1(b), and c′ = f−1(c), we should have c′ = a′ + b′. How to show that a Sugeno λ-measure with λ = 0 is equivalent to a probability measure. Let us consider the auxiliary values A = 1+λ ·a, B = 1+λ ·b, and C = 1+λ ·c. From the formula (2.1), we can now conclude that C = 1+λ ·(a+b+λ ·a ·b) = 1+λ ·a+λ ·b+λ ·a ·b. (2.2) One can easily check that the right-hand side of this formula is equal to the product A ·B of the expressions A = 1+ λ · a and B = 1 + λ · b. Thus, we get