On Uncertainty Analysis Using Random Sets in Science and Engineering - Innovative Management in Information and Production - page 3

At another level when we use natural languages in intelligent systems, the
imprecision is described by fuzzy sets which generalize ordinary sets. We are talking
about random fuzzy sets.

2 The Belief Function Approach to Combining Evidence

Generalizing Bayesian statistics, Dempster ( 1967 )and Shafer ( 1976 ) considered an
approach to quantify evidence by using uncertainty measures under the name of
belief functions. The framework is this.
A “true state of nature” u o is known to be in some finite set U , although it is not
known which element of U is that true state. For each subset A ⊆ U , we express

our “belief” that A contains u o by a number, denoted as F ( A ) . Such a number

F could come from some “evidence.” We are talking about modeling/quantifying

information provided by evidence, i.e., some mathematical theory of evidence.
A belief function on a finite set U is a set function F :2 U

( A )

→ [ 0 , 1 ] such that

(i) F (∅)= 0, F ( U )= 1

2 U , (ii) For any n ≥ 2, and any A 1 , A 2 ,..., A n ∈

n
i = 1 A i ) ≥

∑

) | I | + 1 F F ( ∪ ( − 1 ( ∩ j ∈ I A j ) , (6)

∅ = I ⊆{ 1 , 2 ,..., n }

denotes the cardinality of the set I .
Example [belief functions as distributions of random sets, see Nguyen ( 1978 ,
2012 )]
Let

where | I |

( Ω ,A , P ) be a probability space and ( V ,V ) be an arbitrary measurable

V is called a random element if X − 1 space. A map X : Ω → ( V ) ⊆ A , and its

PX − 1 probability law is the probability measure P X = on V .For U a finite set, and

2 U is called a nonempty random
set whose probability law is completely determined by its distribution function
F :2 U

2 U , being the power set of 2 U , X : V = V Ω →

→ [ 0 , 1 ] ,definedby

F ( A )= P ( X ⊆ A ) . (7)

Now, clearly F (∅)= 0and F ( U )= 1. Moreover, F ( . ) is infinitely monotone.

2 U ,and A i ∈ 2 U , i

Indeed, for B ∈ = 1 , 2 ,..., n ,let

J ( B )= { i : such that B ⊆ A i }. (8)

We h ave

n
i = 1 A i )=

∑
B ⊆∪

∑

F ( ∪ F ( B ) ≥ F ( B ) . (9)

n
i = 1 A i

B ⊆ U , J ( B ) =∅