Download four regular conditional probabilities of fuzzy

CHAPTER
FOUR
REGULAR CONDITIONAL PROBABILITIES OF
FUZZY PROBABILITY SPACES FORMULATED
BY MEANS OF THE FUZZY RELATION ‘LESS
THAN’
Abstract
Chapter 4 deals with the study of regular conditional probabilities
of fuzzy probability spaces. In this chapter as a convenient departure from classical probability space, probability space structured by
means of the fuzzy relation ’less than’ is considered. With the aid
of this new theoretical formulation, the properties of regular conditional probabilities of fuzzy probability spaces are obtained. Baye’s
formula is established in this newly structured probability space.
90
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces 91
4.1
Introduction
Klement et al., [29] have defined a fuzzy space as a pair
(Ω, σ) where Ω denote a fixed crisp set and σ is a fuzzy σ-algebra.
Krzysztof Piasecki [32] has formulated fuzzy probability spaces
defined by means of the fuzzy relation ‘less than’. In [32] fuzzy
probability spaces are defined on the real line and fuzzy σ- algebras of events are introduced in an analogous way to the
classical theory of probability. In [62] Zadeh has defined probability measures for fuzzy events by using the multivalued set
function in the integral. In this chapter as a convenient departure from classical probability spaces Krzysztof Piasecki’s
[32] probability space is considered for obtaining the properties
of regular conditional probabilities of fuzzy probability spaces.
The concept of regular conditional probabilities are introduced
in the Krzysztof Piasecki’s probability space defined in terms
of the fuzzy relation ‘less than’. This kind of probability space
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces 92
is specifically chosen to establish Baye’s theorem. Conditional
probabilities reveal the dependencies between the new and existing evidences. Since Bayesian approach originates from the
ratio form of conditional probability measure, a special class of
fuzzy probability measure is identified for the establishment of
Baye’s theorem and similar results. The notion regular conditional probabilities to the realm of fuzzy random variables and
fuzzy probability measures expose conditions under which the
countably additive requirements can be met.
In Section 4.2, definitions of fuzzy algebra, fuzzy probability measures and fuzzy relation ‘less than’ are introduced. In
Section 4.3 the definitions found in [32] such as fuzzy Borel intervals, fuzzy Borel family, fuzzy random variable etc are introduced. In Section 4.4 the notion of regular conditional probabilities over a fuzzy σ-algebra and regular conditional distribution
function for a fuzzy random variable and its properties are de-
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces 93
rived. In section 4.5 reduced Baye’s formula and properties of
conditional probabilities in the probability space formulated by
the fuzzy relation ‘less than’ are derived.
4.2
Preliminaries and Fuzzy Relation ‘less than’
Definition 4.2.1 ([32]). Let D denotes the family of all constant
functions transforming the set Ω in to [0,1]. Let 0Ω : Ω → {0} and
1Ω : Ω → {1}. A family of functions σ : {µ : Ω → [0, 1]} is known
as a fuzzy σ-algebra if it satisfies the following properties:
∀µ ∈ D, µ ∈ σ
(4.2.1)
∀µ ∈ σ, 1 − µ ∈ σ
(4.2.2)
∀{µn } ∈ σN , sup{µn } ∈ σ
(4.2.3)
n
unless otherwise specifically stated, a fuzzy σ-algebra such that
D = {0Ω , 1Ω } will be considered.
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces 94
Definition 4.2.2. A family of functions σ̂ = {µ : Ω → [0,1]} is
known as a fuzzy algebra if it satisfies the following conditions.
0Ω ∈ σ̂
(4.2.4)
∀µ ∈ σ̂, 1 − µ ∈ σ̂
(4.2.5)
∀(µ, v) ∈ σ̂ × σ̂, µ ∨ v ∈ σ̂
(4.2.6)
Definition 4.2.3. A fuzzy probability measure is a mapping
m : σ → [0, 1] such that
m(0Ω ) = 0
(4.2.7)
m(1Ω ) = 1
(4.2.8)
∀(µ, v) ∈ σ × σ, m(µ ∨ v) + m(µ ∧ v) = m(µ) + m(v)
(4.2.9)
∀{µn } ∈ σN ,
{µn } ↑ µ ⇒ m(µn ) ↑ m(µ)
(4.2.10)
In [29] it is established that in general case a fuzzy probability
measure fails to satisfy the equality
m(1 − µ) = 1 − m(µ)
for all µ ∈ σ.
(4.2.11)
Since (4.2.11) is a necessary condition for the Baye’s formula
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces 95
fuzzy probability measure does not satisfy Baye’s formula. In
this chapter a class of fuzzy probability measures that satisfy
(4.2.11) is considered.
Let X be a given set. A fuzzy relation on X is defined as a
mapping Ψ : X × X → [0, 1]. The inverse and complement of
the fuzzy relation are defined respectively by the membership
functions.
Ψ−1 (x, y) = Ψ(y, x)
and Ψ̄(x, y) = 1 − Ψ(x, y).
A fuzzy relation Ψ is called antireflexive if for each
x ∈ X,
Ψ(x, x) = 0
(4.2.12)
A fuzzy relation Ψ is called antisymmetric if for each pair
(x, y) ∈ X × X
Ψ(x, y) ∧ Ψ(y, x) = 0
(4.2.13)
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces 96
A fuzzy relation Ψ is called transitive if for each pair
(x, y) ∈ X × X.
Ψ(x, y) ≥ sup{Ψ(x, z) ∧ Ψ(z, y)}
(4.2.14)
z∈X
The definition of the fuzzy relation ‘less than’ ρ below leaves
out the properties antireflexivity, antisymmetry and transitivity.
It lays emphasis on connections between inverse relations and
complement relations.
Definition 4.2.4 ([32]). A fuzzy relation ‘less than’ is a mapping
ρ : R̄ × R̄ → [0,1] satisfying the following two conditions.
∀(x, y) ∈ R̄ × R̄, ρ(−x, −y) = ρ(y, x) = ρ−1 (x, y)
≤ ρ̄(x, y) = 1 − ρ(x, y)
∀{yn } ∈ R̄N ,
{yn } ↑ y ⇒ {ρ(., yn )} ↑ ρ(., y),
lim ρ(., y) = 1R̄
y↑∞
(4.2.15)
(4.2.16)
(4.2.17)
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces 97
4.3
Fuzzy Borel Intervals
Definition 4.3.1. A mapping φ[a, b) : R̄ → [0, 1] defined for any
pair (a, b) ∈ R̄ × R̄ by
φ[a, b)(x) = ρ(x, b) ∧ (1 − ρ(x, a))
(4.3.1)
is known as a fuzzy Borel interval.
The smallest fuzzy σ-algebra βρ containing all fuzzy Borel
intervals exists for each fuzzy relation ‘less than’ ρ.
For each n ∈ N, let Nn = {k; k ∈ N; k ≤ n}. For all sequences
{(ak , bk )}k∈Nn such that (ak , bk ) ∈ R̄2 the inequality
a ≥ c and b ≤ d ⇒ φ[a, b) ≤ φ[c, d)
implies
max{φ[ak , bk )} = max{φ[ak , bk )}
k∈Nn
k∈N̄n
where N̄n = {k; k ∈ Nn ; ∃ l ∈ Nn ; ak ≥ al and bk ≤ bl }.
(4.3.2)
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces 98
Definition 4.3.2 ([32]). If β̄ρ = {µ : R̄ → [0, 1]} is the family of all
fuzzy subsets µ such that
µ = max{φ[ak , bk )}
k∈Nn
(4.3.3)
where n is any positive integer and the sequences {ak }k∈Nn and
{bk }k∈Nn are increasing then β̄ρ is known as a fuzzy Borel family.
Let R̄, βρ, m be a fuzzy probability space with a fixed relation
‘less than’.
Definition 4.3.3 ([32]). A fuzzy probability measure
m : βρ → [0, 1] satisfying the following properties is known
as a fuzzy P-measure
∀α ∈ R̄, m(φ[α, α)) = 0
(4.3.4)
∀α ∈ R̄, m(φ[−∞, a) ∨ φ[a, ∞)) = 1
(4.3.5)
Definition 4.3.4 ([32]). Let (Ω, σ, m) be a fuzzy probability space.
A fuzzy random variable is a function χ : Ω → R̄ satisfying the
condition
∀x ∈ R̄,
ρ χ(.), x ∈ σ
(4.3.6)
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces 99
Definition 4.3.5 ([32]). The cumulative distribution function of
the fuzzy random variable χ is a mapping F : R̄ → [0, 1] defined
as
F(x) = m ρ χ(.), x
(4.3.7)
Theorem 4.3.1 ([32]). Each cumulative distribution function
F : R̄ → [0, 1] is a non-decreasing function and continuous from
below. Then
lim F(x) = 1
(4.3.8)
∃α ∈ [0, 1] lim F(x) = α ≥ 0 = F(−∞)
(4.3.9)
x→∞
x→−∞
Definition 4.3.6 ([32]). The mapping ρ(. | v) : σ → [0, 1] defined
for a fuzzy subset v ∈ σ such that m(v) , 0 by the identity
ρ(µ | v) =
m(µ ∧ v)
m(v)
is known as the conditional probability given v.
(4.3.10)
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces100
4.4
Regular Conditional Probabilities
Let G be an arbitrary fuzzy σ-algebra. The conditional probability p(µ | v) is a more intutive object than the conditional
probability p(µ | G). Hence for formal arguments in which G is
an arbitrary fuzzy σ-algebra, we make use of p(µ | G).
If µ1 , µ2 , . . . are disjoint sets in σ, then
∞
∞
X
p ∪ µn | G =
p(µn | G)
n=1
a.e.
n=1
This does not imply that we will be able to choose p(µ | G)(ω),
µ ∈ σ, ω ∈ Ω, so that it is a measure in µ for all ω ∈ Ω. The
difficulty that could emerge under this circumstance is that for
a fixed ω, p(. | G)(ω) need not be countably additive on σ. Thus
there is no guarantee that we can specify the p(µ | G) to be
countably additive in µ. In the following definition suitable
conditions are imposed under which the countable additivity
requirement is realized.
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces101
Definition 4.4.1. Let Y be a fuzzy random variable on (Ω, σ, m)
and G is a fuzzy sub σ-algebra of σ. The function
F = F(ω, y) = m{ρ χ(.), y }(ω) where ω ∈ Ω, y ∈ R is called a
regular conditional distribution function for Y given G iff the
following two conditions are satisfied.
1. For each ω, F(ω, •) is a proper cumulative distribution function of the fuzzy random variable Y, that is increasing and
right continous, with
F(ω, ∞) = m{ρ(χ(•), ∞)}(ω) = 1
and F(ω, −∞) = m{ρ(χ(•), −∞)}(ω) = 0
2. For each y, F(ω, y) = m{ρ(χ(•), y) | G}(ω) for almost every
ω.
Theorem 4.4.1. If Y is a fuzzy random variable on (Ω, σ, m), G a
fuzzy sub σ-algebra of σ, there is always a regular conditional
distribution function for Y given G.
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces102
Proof. Choose a version of Fr (ω) = m{ρ(χ(•), r) | G}(ω) for each
fixed rational r. If r1 , r2 , . . . is a list of the rationals.
Let
Aij = {ω; m ρ(χ(•), r j ) (ω) < m(ρ(χ(•), ri ))(ω)}
A = U{Aij ; ρ(ri , r j )}
Since ρ(ri , r j ) > 0,
n o
m{ρ(χ(•), ri ) | G} ≤ m ρ χ(•), r j | G a.e.
and so m(A) = 0.
Let Bi = {ω; lim m ρ(χ(•), ri + 1/n) (ω) , m ρ(χ(•), ri ) (ω)}
n→∞
B=
∞
S
Bi .
i=1
Since ρ(χ(•), ri + 1/n) ↓ ρ(χ(•), ri ) as n → ∞.
∴
m(ρ(χ(•), ri + 1/n))(ω) → m(ρ(χ(•), ri ))(ω) a.e.
Hence m(B) = 0.
If C = {ω; lim m{ρ(χ(•), n) | G}(ω) , 1},
n→∞
then m(C) = 0.
Similarly D = {ω; lim m{ρ(χ(•), n) , 0} has fuzzy probability
n→−∞
measure zero.
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces103







lim F (ω); if ω < A ∪ B ∪ C ∪ D


r→y+ r
Define F(ω, y) = 






if ω ∈ A ∪ B ∪ C ∪ D.

G(y);
where G(y) is any proper cumulative distribution function of a
fuzzy random variable.
Then F is well defined for if ω < A, Then Fr (ω) is monotone
in r, so that lim+ Fr (ω) exists. Also if ω < A ∪ B then
r→y
lim Fr (ω) = F y (ω) if y is rational. Similarly if ω < A ∪ C ∪ D then
r→y+
lim Fr (ω) = 1 lim Fr (ω) = 0.
r→∞
r→−∞
Fix ω ∈ A ∪ B ∪ C ∪ D. Then F(ω, .) is clearly increasing.
If y < y0 ≤ r then F(ω, y) ≤ F(ω, y0 ) ≤ F(ω, r) = Fr (ω) → F(ω, y)
as r → y.
Thus F(ω, .) is right continuous. If r ≤ y then
F(ω, y) ≥ F(ω, r) = Fr (ω) → 1 as r → ∞. Hence F(ω, y) → 1
as y → ∞. Similarly F(ω, y) → 0 as y → −∞. Hence the first
requirement is satisfied.
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces104
By construction of F we note
m{ρ(χ(•), r) | G}(ω) = Fr (ω) = F(ω, r)
As r ↓ y, F(ω, r) → F(ω, y) for all ω by right continuity and
m{ρ(χ(•), r) | G} ≤ m{ρ χ(•), y | G} a.e.
Thus m{ρ(χ(•), y) | G}(ω) = F(ω, y) for almost every ω. This
establishes the second requirement and the proof is complete.
Definition 4.4.2. Let Y : (Ω, σ) → (Ω0 , σ0 ) be a random object
and G a fuzzy sub σ-field of σ. The function Q : Ω × σ0 → [0, 1]
is called a regular conditional probability for Y given G iff
1. Q(ω, B) is a fuzzy probability measure in B for each fixed
ω ∈ Ω and
2. For each fixed B ∈ σ0 ,
Q(ω, B) = {b ∈ B; m{ρ(χ(.), b)} | G}(ω)
a.e.
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces105
Theorem 4.4.2. Let Y be a fuzzy random variable on (Ω, σ, m), G
a fuzzy sub σ-algebra of σ. There exists a regular conditional
probability for Y given G.
Proof. Let F be a regular conditional distribution function for Y
given G.
Define Q(ω, B) =
R
dF(ω, y).
y∈B
Thus for each ω, Q(ω, .) is the Lebesgue stieltjes measure
corresponding to F(ω, .), hence Q is a fuzzy probability measure
in B if ω is fixed.
Now let C = {B ∈ β̄ρ ; Q(ω, B) = ρ{y ∈ B | G}(ω)
a.e.}
Then C contains all fuzzy borel intervals (−∞, y].
If A, B ∈ C , A ⊂ B then B − A ∈ C . This shows that C contains
all fuzzy borel intervals (a, b], hence all finite disjoint unions
are right semi closed intervals. Thus Q is a regular conditional
probability for Y given G.
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces106
Theorem 4.4.3. Let Y : (Ω, σ) → (Ω0 , σ0 ) be a fuzzy random object and G a fuzzy sub σ-algebra of σ. Suppose there is a map
ψ : (Ω0 , σ0 ) → (R, β̄ρ ) such that ψ is one-to-one, E = ψ(Ω0 ) is a
fuzzy Borel subset of R and ψ−1 is measurable.
Then there is a regular conditional probability for Y given G.
Proof. Let Q0 = Q0 (ω, B), B ∈ β̄ρ , ω ∈ Ω be a regular conditional
probability for the fuzzy random variable ψ(Y) given G.
Define Q(ω, A) = Q0 ω, ψ(A) ; A ∈ σ0 .
Since ψ−1 is measurable ψ(A) ∈ β̄ρ (E) ⊂ β̄ρ and Q is well
defined.
Now Q is a probability measure in A for ω fixed. If A is fixed
then
Q(ω, A) = m{a ∈ ψ(A); ρ ψ(χ(•), a) | G}(ω)
= m{a ∈ A; ρ(χ(•), a) | G}(ω)
This completes the proof.
a.e.
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces107
4.5
Reduced Baye’s Formula
Theorem 4.5.1.
(i) 0 ≤ p(µ | v) ≤ 1
(ii) p µv(1 − µ) | v = 1
(iii) p (µ | v) ∨ (1 − µ | v) = p(µ | v) + p(1 − µ | v).
Proof. (i) We claim 0 ≤
m(µ∧v)
m(v)
≤ 1.
The left side is obvious. For the right side, (µ ∧ v)(x) ≤ v(x).
This shows that m(µ ∧ v) ≤ m(v)
∴
0≤
m(µ∧v)
m(v)
≤1
⇒ 0 ≤ p(µ | v) ≤ 1
(ii) p((µv(1 − µ))/v) =
m(µv(1−µ)∧v)
m(v)
=
m(v)
m(v)
=1
m (µ ∨ (1 − µ) ∧ v)
(iii) p{(µ | v) ∨ (1 − µ) | v} =
m(v)
=
m(µ ∧ v) + m((1 − µ) ∧ v)
m(v)
= p(µ | v) + p(1 − µ | v)
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces108
Definition 4.5.1. Let (Ω, σ, m) be any fuzzy probability space.
The mapping p(. | v) : σ → [0, 1] defined for a fuzzy subset v ∈ σ
such that m(v) , 0 by the identity
p(µ | v) =
m(µ ∧ v)
m(v)
is called the conditional probability given v.
Theorem 4.5.2. For fuzzy subsets µ1 , µ2 , . . . , µn ; µi ∈ σ,
i = 1, 2, . . . , n,
p(µ1 µ2 · · · µn ) = p(µ1 )p(µ2 | µ1 )p(µ3 | µ1 µ2 ) · · · p(µn | µ1 µ2 · · · µn−1 )
provided p(µ1 µ2 · · · µn−1 ) > 0.
Proof. Since p(µ1 ) ≥ p(µ1 µ2 ) · · · ≥ p(µ1 µ2 · · · µn−1 ) > 0 call the
conditional probabilities stated above are well defined.
∴ p(µ1 )p(µ2 | µ1 )p(µ3 | µ1 µ2 ) · · · p(µn | µ1 µ2 · · · µn−1 )
=
m(µ1 ∧ µ2 ∧ · · · ∧ µn )
m(µ1 ) m(µ1 ∧ µ2 ) m(µ1 ∧ µ2 ∧ µ3 )
···
m(Ω) m(µ1 )
m(µ1 ∧ µ2 )
m(µ1 ∧ µ2 · · · µn−1 )
=
m(µ1 ∧ µ2 ∧ · · · ∧ µn )
m(Ω)
= p(µ1 µ2 · · · µn )
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces109
Theorem 4.5.3. If a fuzzy probability measure m satisfies
m(µ ∨ (1 − µ)) = 1
(4.5.1)
m(µ ∧ (1 − µ)) = 0
(4.5.2)
Then for any v ∈ σ,
m(v) = m(µ)p(v | µ) + m(1 − µ)p(v | 1 − µ)
Proof.
v = µ ∨ (1 − µ) ∧ v
= (µ ∧ v) ∨ ((1 − µ) ∧ v)
m(v) = m(µ ∧ v) + m((1 − µ) ∧ v)
m(v) = m(µ)
m(µ ∧ v)
m((1 − µ) ∧ v)
+ m(1 − µ)
m(µ)
m(1 − µ)
m(v) = m(µ)p(v | µ) + m(1 − µ)p(v | 1 − µ)
⇒
m(v) = m(µ)p(v | µ) + m(1 − µ)p(v | 1 − µ)
Ch: 4 Regular Conditional Probabilities of Fuzzy Probability Spaces110
Theorem 4.5.4 (Reduced Baye’s Formula). If a fuzzy probability
measure m satisfies (4.5.1) and (4.5.2) then
p(µ | v) =
m(µ).p(v | µ)
m(µ)p(v | µ) + m(1 − µ)p(v | 1 − µ)
(4.5.3)
for any pair (µ, v) ∈ σ × σ such that m(µ) , 0, m(µ) , 1 and
m(v) , 0.
Proof. m(µ)p(v | µ) + m(1 − µ)p(v | 1 − µ) = m(v)
Then equation (4.5.3) becomes
m(v)p(µ | v) = m(µ)p(v | µ).
This is true because both sides are equal to m(µ ∧ v).