An algorithm for identification of belief measures
Wei Chen
Computer Science,
University of Nebraska at Omaha
wchen@mail.unomaha.edu
Abstract:
Zhengyuan Wang
Mathematics,
University of Nebraska at Omaha
zhenyuanwang@mail.unomaha.edu
usually given by experts or obtained from other
empirical ways. Incompleteness and inconsistence
Identification of a belief measure is to test whether a
containing in the estimation of belief measure are the
measure by given meet the requirement of belief
major problems for its application. As mentioned above,
measure, if contraction exists in given measure, then
belief measure can be computed from a basic
belief measure is generated has small error to given one.
probability assignment for every element in the set we
An algorithm is developed to search for the best fitting
concerned. If we can get all basic probability for every
basic probability assignment based on a given “blank”
element in the universal set, if we can find such a
of belief measure. Through the algorithm, a belief
probability assignment, that belief measure on them has
measure can be identified if it exists. When the exact
no or least error comparing with the estimation of it by
identification fails, a genetic algorithm is activated to
given , we could avoid or reduce incompleteness and
search for a global optimal solution and obtain an
inconsistence which come from subjective views and
approximate optimal belief measure.
other
resources.
Finding
such
basic
probability
assignments is identification of belief measure. By
1. Introduction
identification we use a procedure of fitting an optimal
consistent belief measure to estimation by given, this
Concept of belief measure is introduced in
Dempster-shafer (D-S) theory, which provides a method
optimal belief measure has as low as possible error to
estimation measure from ,for example , experts.
for uncertainty inference and information fusion. The
The belief measure is one of the most important
degree of belief based on belief measure can be got
types of fuzzy measures in common use. From
from basic probability assignment. Different belief
information obtained from the existing experience of
measures can be combined to form a new belief
experts, can be estimation measure of target attributes
measure; this process can be used to combine
sets. However, it does not satisfy, usually, the
information or called information fusion. Through this
requirement being a belief measure. Based on given set
combination, the uncertain on the events can be reduced
function, we may find an approximate optimal belief
[3].
measure through a good search method. There is little
In belief measure, a generalization of classical
research in this process of information uncertainty. In
probability theory that permits the assignment of
our algorithm, we first find out whether there is a
probability masses (beliefs) to all subsets of the
probability assignment meets the requirement. if the
universe and not just to the basic elements.
A measure
estimation of belief measure satisfy the requirement, or
of belief for an assertion is then computed as a
there is no contradiction within the estimation, the
subinterval of [0, 1], where the length of the interval
algorithm
measures the uncertainty.
corresponding probability assignment; if contradiction
In real life, the estimation of belief measure is
will
give
positive
answer,
also
give
exists, a genetic algorithm is activated to search an
approximate solution. In the following section, basic
In our case, let the universal set X be a finite set
concepts and connection of belief measure and
{ x1 , x2 ,..., xn }. The belief measure on E j , j=1, 2,…, s,
probability assignment are given
 ( E j ) are give by human experts subjectively. We
2. Background
need to find 2n  1 values to fit m( Fk ) a basic
probability assignment such that for a belief measure
Belˆ( E j ) built on them contribute least sum of error
2.1 Basic Concept of Belief Measure
compare with the  ( E j ) by given. Ideally, if there
exists a probability assignment contributes no error, this
assignment is the solution without contradiction, the
In this section, we introduce some basic definitions
given measure is identified as a belief measure, if we
and theorem about belief measure
can not find such assignment, and an optimal solution is
search to achieve least error .
Definition 2.1. Let P (P(X)) be the power set of P(X).
If p is a discrete probability measure on (P(X), P (P(X)))
with p( )  0 , then the set of function m: P(X) [0, 1]
determined by
I
3. Identification of belief measure
General structure of identification algorithm is two
any E  P(X)
(2-1)
is called a basic probability assignment o P(X).
part (see figure 1), one is finding contradiction, if
Theorem 2.1. A set function m: P(X) [0, 1] is a basic
contradiction does not exist, the solution without error is
probability assignment if and only if
generated in this part; else a genetic algorithm is
m(E)  P({E})
for
(1) m( )  0 ;
(2)  m( E )  1 ;
activated.
E X
Definition 2.2. If m is a basic probability
3.1 coding the sets
assignment on P(X), then the set function Bel: P(X) 
How the sets present in the algorithm matter a lot. A
[0, 1] determined by
Bel(E)=  m( F ) for any E  P(X)
F E
(2-2)
good coding can make algorithm efficient. A binary
is called a belief measure on (P(X), P (P(X))), or more
string is used in the coding, every bit presents whether
exactly, a belief measure induced from m.
corresponding element in the universe set appear in
Theorem 2.2. If Bel is a belief measure on ( P(X), P
subset, the ith binary bit is 1 in its codes; otherwise the
bit is set as 0. We use bnbn 1...b1 to present the code
(P(X))), then
(BM1)
(BM2)
Bel(  )=0
string. After coding, we transform bit string of the set n
into a real number as indices of the sets, that is
Bel(X)=1;
n
(BM3)
Bel(  Ei ) 
i 1
current set. If the ith element appears in the specific

I {1, 2,..., n}, I  
(1)| I | 1 Bel (  Ei )
i I
where { E1,..., En } is any set finite subclass of P(X);
Theorem 2.3. The basic probability assignment and
n
i   bk 2 n 1 . For example, if the universe set is
k 1
{x1 , x2 , x3 , x4 , x5 , x6 } , a set {x3 , x5 } can be coded as
010100, so it is called set E20 .
its corresponding belief measure satisfy condition:
m( E )   (1)| E  F | Bel ( F ) for any E  (X)
F E
The method of coding helps to find out whether
(2-3)
Ei  E j . The operation is easy to implemented through
Where |E-F| denotes the cardinality number of the set
(E-F).[2], [3]
logic
“or”.
For
example,
sets {x1, x2 , x3}
and
{x2 , x3} are coded as 111 and 110, [(111) or (110)]
2.2 Problem descriptions
=(111),
so
we
{x2 , x3}  {x1, x2 , x3} .
Proposition 3.1
can
get
conclusion
as
 ( E j ) , j=1, 2,s, are given,
Given a universe set X , every element in the
power set P(X) Ei  X ,if using the coding method
Goal function:
s
f  min  ( ( E j )  Belˆ( E j )) 2
above and Ei  E j then i  j
where E j  P(X), j=1, 2,s
Proof: if Ei  E j , the bit strings for Ei and E j are
si
and s j . So there exist at least one element xs
Belˆ( E j ) can be computed from Belˆ( E j )   m( F )
F E j
contained in E j but not Ei ; accordingly, at least one
bit, let say
n
(3-1)
j 1
bs on which is 1 in s j but 0 in si . So
n
i   bk(i ) 2n 1  j   bk( j ) 2n 1 .
k 1
k 1
This statement says the index of any set is greater than
all of its subset. This information can help to locate the
subset of any given set.
A solution is a set of vector with cardinality number
of 2n  1 . We simplify the basic probability assignment
m( Ei ) to mi , i=1, …, 2n  1 . We decode the index i
to
binary
vector
with
dimension
n.
For
example m5  m(101) , the binary vector 101 denotes
Ei
x1 and x3 .Similarly, m6  m(110)
denotes x2 and x3 . Then we will design a genetic
n
n
algorithm for find 2  2 of m( Ei ) , i=1,…, 2  1 , to
is consist of
3.2 Finding contradiction
As mentioned above, if we can find a set of
probability assignment m( Fk ) , k  1... 2n 1 , such that
for a belief measure Belˆ( E j ) built on them contribute
get optimal (3-1).
Step1: randomly generate big population of potential
no sum of error compare with the  ( E j ) by given,
vectors ( m1,..,m2 n 1 ). For every vector, we call it a
j  1...2n 1 , we say the there are no contradict between
chromosome. Every piece of binary number stored in
belief measure. In this situation, an algorithm of
the chromosome denote to corresponding value of
identification could:
2 n  1 , j=1,…P. P is the number of
( j)
pupation. We call vector g i
the ith gene on jth
mi
1) initialize all basic probability assignment blank or
m( Fk )  0 k  1... 2n 1
2n 1 , update all basic probability
assignment with according belief measure. If set Fi
2) From 1 to
( j)
i=1,…
chromosome, mi ( j )  g i ( j ) /  g k ( j ) . All the genes
generated must satisfies the restriction,
2 n 1
g i  [0,1]
is singleton, contains only one element (this is easy
( j)
and  mi  1 . The length of every chromosome
to know when we decode the index of the set), such
depends on the accuracy of every gene on it. In our case,
as
{xs }
,
else m( Fi )   ( Fi ) 
m( Fi )   ( Fi )
then

F j  Fi , j  i
m( F j ) ,
and
,
every gene has same characteristics, it share the same
all
structure and bits. Then using the position of the gene,
m( F j ) , j  i , are available,
3) Check
n
2 1
whether
every
m( Fi )  [0,1]
j 1
we
and
 m( Fi )  1 , if it is true, there is no contradiction
i 1
among belief measure, and the current assignment
is the solution, else there exists contradiction and a
genetic algorithm is activate to search an optimal
solution.
3.3 Search optimal assignment using genetic
algorithm
can decode
every
substring
chromosome into a real value in [0,1] for corresponding
basic probability assignment.
Step 2: According to the goal of optimization, we
choose a criteria for computing the goodness of
chromosomes. For every chromosome, when generated,
can get an evaluation of its goodness using its error to
belief measure given.
s
 r 2   ( ( E j r )  Belˆ( E j r )) 2
(3-2)
Where Belˆ( E j r )   m ( r ) ( F )
(3-3)
j 1
A genetic algorithm is involved in search optimal
solution. We use model this problem in the following:
of binary on
F E j
goodness function:
Gr 
step 5.
m( )
2

2
(3-4)
r
Step 5: control the termination of the search.
Although the GA is good for global searching, there is a
( 1  r  p ). All
situation that the error of best chromosome can not be
chromosomes can get Gr  [01] and  r 2 . We sort all
chromosome behaves. If it changes smaller and smaller,
chromosomes on their goodness in descending order. So
In the end, we decode the chromosome to the vector
m( 2 )
Where
is min
r2
the best result of this generation among the population
convergent to 0, we can observe how the error of best
then we can exit the search and report the chromosome.
we need. This vector is the optimal belief measure.
are list on the top, the worst cases are on the button.
4. Simulations and conclusions
Step 3: generate new chromosomes on randomly
chosen parents. A random number generator is used in
this step. For every chromosome with Gr , we can assign
4.1 Some simulations
a probability value to it as
In real world, the belief measure is given subjectively
Pr 
Gr
by experts or other empirical ways. We give sets of
, r=1, 2,p
(3-5)
Gi
We usei
random
number generator to get a value
1
known estimation of belief measures and then use our
from 0 to 1. This number will fall into an interval
[  Pj ,  Pj , ], j is the parent chosen. (Let P0  0 and j=1,
listed in the Table 1 and 2.
Setting situation as: universe set X  {x1, x2 , x3} ; in
2,…,p). The generations of new chromosome have 3
case 1 and 2, belief measures are given as in table 1 and
ways, crossover, mutation and realignment. All of them
2, the results of identification shows in the tables.
j 1
j
i 0
i 0
p
algorithm to get optimal solutions. The comparisons are
are two points’ operators. The crossover means we
The results shows the proposed identification
randomly choose two positions on the target pair of
algorithm can not only find optimal solution, but also
chromosomes, the string between two points get
can find whether there is contradiction among in
exchanged into each other. (See figure 2). The mutation
estimation of belief measures by given.
is choosing two positions randomly on the target pair of
chromosomes and restore the opposite values in
4.2 Conclusions
corresponding position, such as change 1 to 0 and 0 to 1.
(See figure 3). The realignment is choosing two
The identification algorithm proposed in the paper is
positions randomly on the target pair of chromosomes,
used to get a set of probability assignments by
the chromosomes become 3 substrings for each, and we
subjective belief measures given. The coding method is
relocate these 3 substrings to get a new chromosome.
used to present the power set, which is embedded in the
(See figure 4).
Step 4: examine the new chromosomes and update
algorithm and helps to deal with relationship between
the population. As mentioned above, every time a pair
part is checking existence of contradiction, the second
of new chromosomes generate. For each chromosome,
part is genetic algorithm activated to find optimal
corresponding error and goodness function is computed.
solution if contradiction exists. Without contradiction,
We use these values to resort the population. If new
the algorithm is very fast, but genetic algorithm search
chromosome is better than the worst one on the button
takes time. The reason to use it is that genetic algorithm
of the list, it will insert into the list, otherwise the new
is global search, if time is enough, the solution found is
chromosome is removed. If we can get an error of
global optimal.
the sets. The algorithm is composed of two parts; first
chromosome equal to 0, then we can direct report this
Due to the rareness of previous work having done in
chromosome and exit the searching. Otherwise go to the
this field, there are further improvement can be done, a
lot of search methods and soft computing methods can

Set
be used.
When the problem of identification of belief
measure is solved, the belief measure get more abroad
application in information fusion, decision making
and other fields related to uncertainty management.
{x1}
{x2}
{x1, x2}
{x3}
{x1, x3}
{x2, x3}
{x1, x2,
x3}
given
by
Probability
by cmp
Bel
cmp
0.1
0.2
0.2
0.3
0.5
0.9
1
0.2
0.1
0.2
0.3
0.05
0.1
0.05
0.1
0.1
0.2
0.3
0.05
0.1
1.0
by
contradictio
n
No
Error=0
Table 1. In case 1, the belief measures are given without
contradiction, the result of identification algorithm shows
error is 0.

Set
Figure 1. Structure of identification algorithm
Probability by
cmp
Bel by cmp
contradiction
given
{x1}
0.2
0.069745
0.0697
yes
{x2}
0.1
0.16503
0.1650
{x1, x2}
0.5
0
0.2348
{x3}
0.3
0.33399
0.3340
{x1, x3}
0.55
0.061886
0.4656
{x2, x3}
0.5
0.36935
0.8684
{x1,
x3}
1
0
1.0000
x2,
by
Error=0.0067
Table 2. In case 2, the belief measures are given with
contradiction, the result of identification algorithm shows
error is small
Reference
[1] Z. Wang and G. J. Klir, Fuzzy Measure Theory,
Plenum Press, New York, 1992.
Figure 2. Crossover
[2] S. T. Wierzchon, “An algorithm for identification of
fuzzy measure”, Fuzzy Sets and Systems 9 (1983) 69-78.
[3] G. Shafer, A Mathematical Theory of Evidence,
Princeton University Press, Princeton (1976)
Figure 3. Mutation
Figure 4. Realignment