32nd International Conference on Mathematical Methods in Economics 2014
Probability of fuzzy events
Ondřej Pavlačka1, Pavla Rotterová2
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
1
Introduction
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, [email protected]
2 Palacký University Olomouc, Faculty of Science, Department of Mathematical Analysis and Applications of Mathematics, 17. listopadu 12, 771 46 Olomouc, Czech Republic, [email protected]
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32nd International Conference on Mathematical Methods in Economics 2014
1.
p(Ω) = 1,
(1)
2. for any A1 , A2 , . . . ∈ A such that Ai ∩ Aj = ∅ for any i, j ∈ N, i 6= j:
!
∞
∞
[
X
p
An =
p(An ).
n=1
(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(Ac ) = 1 − p(A),
5. P
for any A1 , . . . , An ∈ A such that Ai ∩ Aj = ∅ for all i, j ∈ {1, . . . , n}, i 6= j, and
n
i=1 p(Ai ) = 1.
Sn
i=1
Ai = Ω:
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 µA : 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 | µA (ω) = 1}, and a support of A, i.e. Supp A := {ω ∈ Ω | µA (ω) > 0}, respectively.
For any α ∈ (0, 1], Aα means an α-cut of A, i.e. Aα := {ω ∈ Ω | µA (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 µA coincides in such a case with the characteristic function χA of A. For a crisp set A, Supp A =
A, and Aα = 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 µA (ω) ≤ µB (ω)
holds for all ω ∈ Ω. Obviously, A ⊆ B if and only if Aα ⊆ Bα 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 µA∩B (ω) = min{µA (ω), µB (ω)} and
µA∪B (ω) = max{µA (ω), µB (ω)}, respectively. Since the minimum and maximum operations are used,
(A ∩ B)α = Aα ∩ Bα and (A ∪ B)α = Aα ∪ Bα hold for all α ∈ (0, 1].
A complement of a fuzzy set A is a fuzzy set Ac whose membership function is for all ω ∈ Ω given by
µAc (ω) = 1 − µA (ω). Obviously, (Ac )α = (Aα )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 = Rn and A = Bn , where Bn denotes the
σ-algebra of Borel sets in Rn , 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. Aα ∈ A for any α ∈ (0, 1]. The family of all such fuzzy events, let us
denote it by AF , forms a σ-algebra of fuzzy sets on Ω (see Negoita and Ralescu [3]), i.e.
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32nd International Conference on Mathematical Methods in Economics 2014
1. Ω ∈ AF ,
2. for any A1 , A2 . . . ∈ AF :
S∞
n=1
An ∈ AF ,
3. for any A ∈ AF : Ac ∈ AF ,
Note that A ⊆ AF .
Now, the probability measure p needs to be extended to the case of fuzzy event. Let us denote the
extension by pF . Zadeh [8] defined the probability pF (A) of a fuzzy event A ∈ AF as the expected value
of its membership function µA , i.e. by the Lebesgue-Stieltjes integral
Z
pF (A) := E(µA ) =
µA (ω)dp.
(3)
Ω
The existence of the above Lebesgue-Stieltjes integral follows directly from the assumption that µA is
A-measurable. It can be easily seen that for any crisp random event A ∈ A, pF (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:
Z
1
pF (A) =
p(Aα )dα.
0
It is obvious from (3) that pF : AF → [0, 1]. Furthermore, Negoita and Ralescu [3] showed that pF
fulfills also the two conditions for a probability measure,
1, and for any A1 , A2 , . . . ∈ AF
S∞ i.e. pF (Ω)P=
∞
such that Ai ∩ Aj = ∅ for any i, j ∈ N, i 6= j: pF ( n=1 An ) = n=1 pF (An ). As also AF forms a
σ-algebra of fuzzy sets on Ω, they called the ordered triple (Ω, AF , pF ) a fuzzy probability space.
Let us show now that pF possesses also the other properties of the probability measure mentioned
in Section 2. Obviously, pF (∅) = p(∅) = 0. Zadeh [8] showed that if A, B ∈ AF , A ⊆ B, then
pF (A) ≤ pF (B), and that for all A, B ∈ AF , pF (A ∪ B) = pF (A) + pF (B) − pF (A ∩ B). Furthermore,
for any A ∈ AF :
pF (Ac ) = E(µAc ) = E(1 − µA ) = 1 − E(µA ) = 1 − pF (A).
The fifth property can be generalized as follows:
Proposition 1. If A1 , . . . , An ∈ AF form a fuzzy partition of Ω, i.e.
then
n
X
pF (Ai ) = 1.
Pn
i=1
µAi (ω) = 1 for all ω ∈ Ω,
(4)
i=1
Proof.
n
X
i=1
pF (Ai ) =
n
X
E(µAi ) = E
i=1
n
X
!
µAi
= E(1) = 1,
i=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 pF given by (3) represents
a correct extension of the probability measure p. However, the question is if for a fuzzy event A ∈ AF
which is not crisp, the value pF (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 pF (A) = 0.5, the only thing we know is that we can expect in the future the interest rate i
% p.a. such that µA (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
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32nd International Conference on Mathematical Methods in Economics 2014
not related to the chance that A will occur in the future. In fact, if an interest rate i∗ % p.a. such that
µA (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 pF (A)
would have the same meaning as the probability of a crisp event is that the membership degree µA (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 pF (A) says nothing about the fuzziness of a fuzzy event A ∈ AF .
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 b1 , b2 , . . . , b20 of various sizes. What is the probability that a ball drawn at random is large?”
A discrete fuzzy set of large balls BL is given by the following formula:
BL = 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 ,
where elements of the set are in the form
µBL (bi )
(5)
|bi , i = 1, . . . , 20.
According to the formula (3), a probability of a large ball drawing is obtained as follows:
20
pF (BL ) =
1
1 X
µB (bi ) =
· 9.8 = 0.49.
20 i=1 L
20
(6)
However, the result expresses the expected value of the randomly drawn ball membership degree to the
fuzzy set BL . Does this value really express a probability that a ball drawn at random is large? If e.g.
the ball b1 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 AF denotes the family of all fuzzy events introduced
in Section 4. The fuzzy probabilities of fuzzy events are assigned by a mapping PF : AF → F([0, 1])
that is defined in the following way: For any fuzzy event A ∈ AF , the membership function of the fuzzy
probability PF (A) is given for all p̂ ∈ [0, 1] as follows:
(
sup{α ∈ (0, 1] | p̂ = p(Aα )}, if {α ∈ (0, 1] | p̂ = p(Aα )} 6= ∅,
µPF (A) (p̂) =
(7)
0,
otherwise.
The membership function µPF (A) is interpreted as a possibility distribution, i.e. the value µPF (A) (p̂)
means the degree of possibility that the probability of a fuzzy event A is equal to p̂. For illustration, the
membership function of the fuzzy probability PF (BL ) of the fuzzy event BL that was defined in Example
1 is depicted in Fig. 1.
Let us examine now the properties of PF . It can be easily seen from (7) that for any crisp event A ∈ A,
the fuzzy probability PF (A) ”coincides” with the value p(A); µPF (A) (p(A)) = 1, and µPF (A) (p̂) = 0 for
any p̂ 6= p(A). Hence,
(
1, if p̂ = 1,
µPF (Ω) (p̂) =
0, otherwise,
i.e. the first property of a probability measure given by (1) can be observed also for PF .
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
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32nd International Conference on Mathematical Methods in Economics 2014
Figure 1 Membership function of the fuzzy probability PF (BL ).
and difference. The extension is not based on the well known extension principle proposed by Zadeh [7].
For any A, B ∈ AF , the membership functions of the fuzzy sets PF (A) ⊕ PF (B) and PF (A) PF (B) are
defined for all p̂ ∈ R in the following way:
(
sup{α ∈ (0, 1] | p̂ = p(Aα ) + p(Bα )}, if {α ∈ (0, 1] | p̂ = p(Aα ) + p(Bα )} 6= ∅,
µPF (A)⊕PF (B) (p̂) =
0,
otherwise,
(
sup{α ∈ (0, 1] | p̂ = p(Aα ) − p(Bα )}, if {α ∈ (0, 1] | p̂ = p(Aα ) − p(Bα )} 6= ∅,
µPF (A)PF (B) (p̂) =
0,
otherwise.
The second fundamental property of a probability measure given by (2) can be observed for PF in
the following way:
Proposition 2. For any A1 , A2 , . . . ∈ AF such that Ai ∩ Aj = ∅ for any i, j ∈ N, i 6= j:
!
∞
[
PF
Ai = PF (A1 ) ⊕ PF (A2 ) ⊕ . . . .
(8)
i=1
Proof. The assumptions imply that Aiα ∩ Ajα = ∅ for any i, j ∈ N, i 6= j and for all α ∈ (0, 1]. Then, for
any p̂ ∈ [0, 1]:
(
S∞
S∞
sup{α ∈ (0, 1] | p̂ = p(( i=1 Ai )α )}, if {α ∈ (0, 1] | p̂ = p(( i=1 Ai )α )} 6= ∅,
S
µPF ( ∞
(p̂) =
i=1 Ai )
0,
otherwise.
(
S∞
S∞
sup{α ∈ (0, 1] | p̂ = p( i=1 Aiα )}, if {α ∈ (0, 1] | p̂ = p( i=1 Aiα )} 6= ∅,
=
0,
otherwise.
(
P∞
P∞
sup{α ∈ (0, 1] | p̂ = i=1 p(Aiα )}, if {α ∈ (0, 1] | p̂ = i=1 p(Aiα )} 6= ∅,
=
0,
otherwise.
= µPF (A1 )⊕PF (A2 )⊕... (p̂).
As for the other properties recalled in Section 2, only the first three can be observed for PF : The fuzzy
probability PF (∅) coincides with zero, since µPF (∅) (0) = 1, and µPF (∅) (p̂) = 0 for any p̂ 6= 0. If A, B ∈ AF ,
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32nd International Conference on Mathematical Methods in Economics 2014
A ⊆ B, then for all α ∈ (0, 1], min{p̂ | p̂ ∈ PF (A)α } ≤ min{p̂ | p̂ ∈ PF (B)α }, and max{p̂ | p̂ ∈ PF (A)α } ≤
max{p̂ | p̂ ∈ PF (B)α }. These inequalities follow from the fact that p(Aα ) ≤ p(Bα ) for all α ∈ (0, 1]. Thus,
we can conclude that PF (A) ≤ PF (B). For any A, B ∈ AF , PF (A ∪ B) = PF (A) ⊕ PF (B) PF (A ∩ B).
It follows from the fact that for all α ∈ (0, 1]: p((A ∪ B)α ) = p(Aα ∪ Bα ) = p(Aα ) + p(Bα ) − p(Aα ∩ Bα ) =
p(Aα ) + p(Bα ) − p((A ∩ B)α ). The last two properties of a probability measure are not retained in the
case of fuzzy events.
The fuzzy probability PF 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 pF , PF has the common probabilistic interpretation; from the
fuzzy probability PF (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, PF 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.
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[4] Talašová, J. and Pavlačka, O.: Fuzzy Probability Spaces and Their Applications in Decision Making.
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[6] Zadeh, L. A.: Fuzzy Probabilities. Information Processing & Management 20 (1974), 363–372.
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