Probability of fuzzy events

Probability of fuzzy events 1 Introduction Ondřej Pavlačka 1, Pavla Rotterová 2 Abstract. In economic practice, we often deal with events that are defined only vaguely. Such indeterminate events can be modelled by fuzzy sets. In the paper, we examine two main ways of expressing a probability of fuzzy events that are proposed in the literature. We study their mathematical properties and discuss their interpretation. We conclude that none of the approaches gives an appropriate probability of a fuzzy event, and thus, that the question How should the probability of a fuzzy event be expressed? is still an open problem. Keywords: fuzzy probability spaces, fuzzy events, probability measure, decision making under risk. JEL classification: C44 AMS classification: 90B50 In the models of decision making under risk, a probability space is considered, i.e. we are able to assign probabilities to some precisely defined random events, like an interest rate is less than 1.5 % p.a., a loss is greater than 100 000 CZK, etc. However, in economic practice we often deal with events that are defined only vaguely, like a low interest rate, an appropriate revenue, a big loss, etc. Such indeterminate events can be adequately modelled by fuzzy sets on the universal set (see [7]). As we often need to estimate the probability of such events, we need to extend the given probability space to the case of fuzzy events. Extending the given probability space to the case of fuzzy events means: first, to determine which fuzzy sets on the corresponding universal set are eligible to be fuzzy events, and consequently, to define the way of expressing probabilities of such fuzzy events. In the literature, fuzzy events are typically defined as the fuzzy sets whose α-cuts are random events. Two main approaches to expressing their probabilities were established. The common way was introduced by Zadeh [8] in 1968. He defined the probability of a fuzzy event as the expected value of its membership function, i.e. the probability of a fuzzy event is a real number from the unit interval. Another way, proposed by Yager [5] in 1979 and independently by Talašová and Pavlačka [4] in 2006, consists in expressing the probability of a fuzzy event by a fuzzy probability - a fuzzy set defined on [0,1] whose membership function is derived from the probabilities of the α-cuts of the fuzzy event. The aim of the paper is to examine if both the approaches are appropriate to be used in practice. The paper is organized as follows. In Section 2, a definition of a probability space and some important properties of a probability measure are recalled. Section 3 is devoted to basic notions from fuzzy sets theory. In the next two sections, we examine the two different approaches to expressing the probability of fuzzy events. Finally, some concluding remarks are given in Section 6. 2 A probability space and the properties of a probability measure A probability space, introduced by Kolmogorov [2] in 1933, is an ordered triple (Ω, A, p), where Ω denotes a non-empty set of all elementary events (future states of the world), A represents the set of all considered random events (A forms a σ-algebra of subsets of Ω), and p : A [0, 1] is a probability measure that assigns to each random event A A its probability p(a) [0, 1] satisfying the following conditions: 1 Palacký University Olomouc, Faculty of Science, Department of Mathematical Analysis and Applications of Mathematics, 17. listopadu 12, 771 46 Olomouc, Czech Republic, ondrej.pavlacka@upol.cz 2 Palacký University Olomouc, Faculty of Science, Department of Mathematical Analysis and Applications of Mathematics, 17. listopadu 12, 771 46 Olomouc, Czech Republic, pavla.melicherikova01@upol.cz 760

1. p(ω) = 1, (1) 2. for any A 1, A 2,... A such that A i A j = for any i, j N, i j: ( ) p A n = p(a n ). (2) n=1 Any probability measure p possesses also the following well known properties that play a significant role in practical applications: 1. p( ) = 0, 2. for all A, B A, A B: p(a) p(b), 3. for all A, B A: p(a B) = p(a) + p(b) p(a B), 4. for any A A: p(a c ) = 1 p(a), 5. for any A 1,..., A n A such that A i A j = for all i, j 1,..., n}, i j, and n A i = Ω: n p(a i) = 1. n=1 Later in the paper, we will study retaining of these properties also for the probabilities of fuzzy events. 3 Fuzzy sets In this section, let us briefly introduce basic notions of fuzzy sets theory. We will recall the definitions of a fuzzy set and its characteristics, inclusion between fuzzy sets, and basic operations with fuzzy sets. A fuzzy set A on a nonempty set Ω is characterized by its membership function µ A : Ω [0, 1]. The family of all fuzzy sets on Ω will be denoted by F(Ω). By Core A and Supp A, we denote a core of A, i.e. Core A := ω Ω µ A (ω) = 1}, and a support of A, i.e. Supp A := ω Ω µ A (ω) > 0}, respectively. For any α (0, 1], A α means an α-cut of A, i.e. A α := ω Ω µ A (ω) α}. Let us note that any crisp set A Ω can be viewed as a fuzzy set of a special kind; the membership function µ A coincides in such a case with the characteristic function χ A of A. For a crisp set A, SuppA = A, and A α = A for all α (0, 1]. A fuzzy set A F(Ω) is said to be a subset of B F(Ω), we will denote it by A B, if µ A (ω) µ B (ω) holds for all ω Ω. Obviously, A B if and only if A α B α for any α (0, 1]. The intersection and union of two fuzzy sets A, B F(Ω) are defined as fuzzy sets A B, A B F(Ω) whose membership functions are for all ω Ω given by µ A B (ω) = minµ A (ω), µ B (ω)} and µ A B (ω) = maxµ A (ω), µ B (ω)}, respectively. Since the minimum and maximum operations are used, (A B) α = A α B α and (A B) α = A α B α hold for all α (0, 1]. A complement of a fuzzy set A is a fuzzy set A c whose membership function is for all ω Ω given by µ A c(ω) = 1 µ A (ω). Obviously, (A c ) α = (A α ) c for any α (0, 1]. 4 Extension of a given probability space to the case of fuzzy events Now, let us assume that a probability space (Ω, A, p) is given and needs to be extended to the case of fuzzy events defined on Ω. In this section, we will analyse the most common way of such extension that was proposed by Zadeh [8] (in fact, Zadeh [8] considered Ω = R n and A = B n, where B n denotes the σ-algebra of Borel sets in R n, but the proposed extension can be without any modifications applied to the case of general Ω and general A). As it was mentioned in Introduction, first of all, we have to determine which fuzzy sets on Ω are eligible to be fuzzy events. According to Zadeh [8], a fuzzy event is a fuzzy set A F(Ω) whose membership function is A-measurable, i.e. A α A for any α (0, 1]. The family of all such fuzzy events, let us denote it by A F, forms a σ-algebra of fuzzy sets on Ω (see Negoita and Ralescu [3]), i.e. 761

1. Ω A F, 2. for any A 1, A 2... A F : n=1 A n A F, 3. for any A A F : A c A F, Note that A A F. Now, the probability measure p needs to be extended to the case of fuzzy event. Let us denote the extension by p F. Zadeh [8] defined the probability p F (A) of a fuzzy event A A F as the expected value of its membership function µ A, i.e. by the Lebesgue-Stieltjes integral p F (A) := E(µ A ) = µ A (ω)dp. (3) The existence of the above Lebesgue-Stieltjes integral follows directly from the assumption that µ A is A-measurable. It can be easily seen that for any crisp random event A A, p F (A) = p(a). Talašová and Pavlačka [4] showed that the probability of a fuzzy event A given by (3) can be equivalently expressed as follows: p F (A) = 1 0 Ω p(a α )dα. It is obvious from (3) that p F : A F [0, 1]. Furthermore, Negoita and Ralescu [3] showed that p F fulfills also the two conditions for a probability measure, i.e. p F (Ω) = 1, and for any A 1, A 2,... A F such that A i A j = for any i, j N, i j: p F ( n=1 A n) = n=1 p F (A n ). As also A F forms a σ-algebra of fuzzy sets on Ω, they called the ordered triple (Ω, A F, p F ) a fuzzy probability space. Let us show now that p F possesses also the other properties of the probability measure mentioned in Section 2. Obviously, p F ( ) = p( ) = 0. Zadeh [8] showed that if A, B A F, A B, then p F (A) p F (B), and that for all A, B A F, p F (A B) = p F (A) + p F (B) p F (A B). Furthermore, for any A A F : p F (A c ) = E(µ A c) = E(1 µ A ) = 1 E(µ A ) = 1 p F (A). The fifth property can be generalized as follows: Proposition 1. If A 1,..., A n A F then Proof. n p F (A i ) = form a fuzzy partition of Ω, i.e. n µ A i (ω) = 1 for all ω Ω, n p F (A i ) = 1. (4) ( n n ) E(µ Ai ) = E µ Ai = E(1) = 1, where 1 is a random variable on Ω such that 1(ω) = 1 for all ω Ω. Thus, we can see that from a mathematical point of view, the mapping p F given by (3) represents a correct extension of the probability measure p. However, the question is if for a fuzzy event A A F which is not crisp, the value p F (A) represents its probability in a common sense. Let us discuss now the problem in more detail. The probability p(a) of a uniquely determined crisp event A is commonly interpreted as a measure of the chance that the event A occurs in the future. For instance, if A denotes an event the interest rate will be less or equal to 2 % p.a. and p(a) = 0.5, then we know that there is the exactly same chance that the interest rate will be less or equal to 2 % p.a. or that it will be greater than 2 % p.a. In other words, if we have the possibility to infinitely many times repeat the process, we can expect that the event A will occur in 50 % cases. However, if we have a vaguely defined event the interest rate will be low, expressed by a fuzzy set A, and if p F (A) = 0.5, the only thing we know is that we can expect in the future the interest rate i % p.a. such that µ A (i) = 0.5, i.e. the expected possibility that i is small is equal to 0.5. And this is the completely different meaning than the common interpretation of the probability of a crisp event; it is 762

not related to the chance that A will occur in the future. In fact, if an interest rate i % p.a. such that µ A (i ) (0, 1) will occur in the future, we will not be able even to uniquely decide whether A occurred or not, i.e. whether the interest rate i is or is not small. Let us note that the only way the value p F (A) would have the same meaning as the probability of a crisp event is that the membership degree µ A (u) is interpreted as the probability that an object u belongs to A. Such interpretation of the membership degrees was proposed by Hisdal [1]. Another problem is that the value p F (A) says nothing about the fuzziness of a fuzzy event A A F. This is the reason why Yager [5], and Talašová and Pavlačka [4] proposed the idea that a probability of a fuzzy event should be also fuzzy. This approach will be analysed in the next section. Example 1. Let us consider the following example (its idea is taken from Zadeh [6]): An urn contains 20 balls b 1, b 2,..., b 20 of various sizes. What is the probability that a ball drawn at random is large? A discrete fuzzy set of large balls B L is given by the following formula: B L = 0.5 b1, 0 b2, 1 b3, 0.2 b4, 1 b5, 0.7 b6, 0.3 b7, 0 b8, 1 b9, 0.4 b10, 1 b11, 0.6 b12, 0.1 b13, 0.8 b14, 1 b15, 0 b16, 0.9 b17, 0.3 b18, 0 b19, 0 b20 }, (5) where elements of the set are in the form µ B L (b i) bi, i = 1,..., 20. According to the formula (3), a probability of a large ball drawing is obtained as follows: p F (B L ) = 1 20 20 µ BL (b i ) = 1 9.8 = 0.49. (6) 20 However, the result expresses the expected value of the randomly drawn ball membership degree to the fuzzy set B L. Does this value really express a probability that a ball drawn at random is large? If e.g. the ball b 1 that is a large ball with a membership degree 0.5 will be drawn, are we able to say whether the event a ball drawn at random is large occurred? 5 Fuzzy probabilities of fuzzy events Let us focus now on another way of expressing the probabilities of fuzzy events - by so called fuzzy probabilities whose membership functions are derived from the probabilities of α-cuts of fuzzy events. This means the resulting fuzzy probabilities reflect the fuzziness of the fuzzy events. The approach that will be described further was introduced by Yager [5] and by Talašová and Pavlačka [4]. In fact, the similar but not the same idea was proposed earlier by Zadeh [6]. Let (Ω, A, p) is a given probability space, and let A F denotes the family of all fuzzy events introduced in Section 4. The fuzzy probabilities of fuzzy events are assigned by a mapping P F : A F F([0, 1]) that is defined in the following way: For any fuzzy event A A F, the membership function of the fuzzy probability P F (A) is given for all ˆp [0, 1] as follows: µ PF (A)(ˆp) = supα (0, 1] ˆp = p(a α )}, if α (0, 1] ˆp = p(a α )}, The membership function µ PF (A) is interpreted as a possibility distribution, i.e. the value µ PF (A)(ˆp) means the degree of possibility that the probability of a fuzzy event A is equal to ˆp. For illustration, the membership function of the fuzzy probability P F (B L ) of the fuzzy event B L that was defined in Example 1 is depicted in Fig. 1. Let us examine now the properties of P F. It can be easily seen from (7) that for any crisp event A A, the fuzzy probability P F (A) coincides with the value p(a); µ PF (A)(p(A)) = 1, and µ PF (A)(ˆp) = 0 for any ˆp p(a). Hence, 1, if ˆp = 1, µ PF (Ω)(ˆp) = 0, otherwise, i.e. the first property of a probability measure given by (1) can be observed also for P F. For showing the retaining of some other properties, it is convenient to introduce the following special operations and with fuzzy probabilities that represent an extension of the arithmetic operations sum (7) 763

Figure 1 Membership function of the fuzzy probability P F (B L ). and difference. The extension is not based on the well known extension principle proposed by Zadeh [7]. For any A, B A F, the membership functions of the fuzzy sets P F (A) P F (B) and P F (A) P F (B) are defined for all ˆp R in the following way: µ PF (A) P F (B)(ˆp) = µ PF (A) P F (B)(ˆp) = supα (0, 1] ˆp = p(a α ) + p(b α )}, if α (0, 1] ˆp = p(a α ) + p(b α )}, 0, otherwise, supα (0, 1] ˆp = p(a α ) p(b α )}, if α (0, 1] ˆp = p(a α ) p(b α )}, The second fundamental property of a probability measure given by (2) can be observed for P F the following way: in Proposition 2. For any A 1, A 2,... A F such that A i A j = for any i, j N, i j: ( ) P F A i = P F (A 1 ) P F (A 2 ).... (8) Proof. The assumptions imply that A iα A jα = for any i, j N, i j and for all α (0, 1]. Then, for any ˆp [0, 1]: µ PF ( supα (0, 1] ˆp = p(( Ai) (ˆp) = A i) α )}, if α (0, 1] ˆp = p(( A i) α )}, = = supα (0, 1] ˆp = p( A iα)}, if α (0, 1] ˆp = p( A iα)}, supα (0, 1] ˆp = p(a iα)}, if α (0, 1] ˆp = p(a iα)}, = µ PF (A 1) P F (A 2)...(ˆp). As for the other properties recalled in Section 2, only the first three can be observed for P F : The fuzzy probability P F ( ) coincides with zero, since µ PF ( )(0) = 1, and µ PF ( )(ˆp) = 0 for any ˆp 0. If A, B A F, 764

A B, then for all α (0, 1], minˆp ˆp P F (A) α } minˆp ˆp P F (B) α }, and maxˆp ˆp P F (A) α } maxˆp ˆp P F (B) α }. These inequalities follow from the fact that p(a α ) p(b α ) for all α (0, 1]. Thus, we can conclude that P F (A) P F (B). For any A, B A F, P F (A B) = P F (A) P F (B) P F (A B). It follows from the fact that for all α (0, 1]: p((a B) α ) = p(a α B α ) = p(a α )+p(b α ) p(a α B α ) = p(a α ) + p(b α ) p((a B) α ). The last two properties of a probability measure are not retained in the case of fuzzy events. The fuzzy probability P F whose membership function is given by (7) seems to be not fully appropriate to be used in practice. Its main advantage consists in the fact that it reflects the fuzziness of fuzzy events. Another advantage is that in contrast to p F, P F has the common probabilistic interpretation; from the fuzzy probability P F (A), we can see the probabilities of occurring the particular α-cuts of A which are crisp events. However, construction of fuzzy probabilities and subsequent calculation with them are not simple, particularly in the discrete case. Moreover, P F does not retain some of the important properties of a probability measure. 6 Conclusion We have examined two ways of expressing the probability of fuzzy events that are proposed in the literature. The first, most common way consists in expression the probability of a fuzzy event as the expected value of its membership function. We have shown that this approach has a nice mathematical properties, but lacks the common interpretation of a probability and does not reflect the fuzziness of fuzzy events. The second way is based on the idea that the probability of a fuzzy event should be also fuzzy. The membership function of the resulting fuzzy probability is derived from the probabilities of α-cuts of fuzzy events. However, the fuzzy probabilities do not have some important properties of a probability measure and the calculations with them could be awkward. Thus, we have shown that none of the approaches gives an appropriate probability of a fuzzy event. The question How should the probability of a fuzzy event be expressed? is still an open problem. Acknowledgements The research is supported by the grant No. GA 14-02424S of the Grant Agency of the Czech Republic and by the grant IGA PrF 2014028 Mathematical Models of the Internal Grant Agency of the Palacký University in Olomouc. References [1] Hisdal, E.: Are grades of membership probabilities? Fuzzy Sets and Systems 25 (3) (1988), 325 348. [2] Kolmogorov, A.: Grundbegriffe der Wahrscheinlichkeitsrechnung. Julius Springer, Berlin, 1933. [3] Negoita, C. V. and Ralescu, D.: Applications of Fuzzy Sets to System Analysis (1st ed.). Birkhuser Verlag Edituria Technica, Stuttgart, 1975. [4] Talašová, J. and Pavlačka, O.: Fuzzy Probability Spaces and Their Applications in Decision Making. Austrian Journal of Statistics 35 (2&3) (2006), 347 356. [5] Yager, R.R.: A note on probabilities of fuzzy events. Information Sciences 18 (2) (1979), 113-129. [6] Zadeh, L. A.: Fuzzy Probabilities. Information Processing & Management 20 (1974), 363 372. [7] Zadeh, L. A.: Fuzzy Sets. Information and Control 8 (1965), 338 353. [8] Zadeh, L. A.: Probability Measures of Fuzzy Events. Journal of Mathematical Analysis and Applications 23 (1968), 421 427. 765