1. Dempster-Shafer Theory
Fundamentals of
Dempster-Shafer Theory
College of Computing and Informatics
University of North Carolina, Charlotte
presented by
Zbigniew W. Ras
University of North Carolina, Charlotte, NC
Warsaw University of Technology, Warsaw, Poland 1

2. Examples – Datasets (Information Systems)

3. Information System How to interpret term “blond”?
Color_of_Hair
Nationality X1
Blond X2
Spanish X3
Black X4
German
How to interpret term “blond”?
How to interpret “blond + Spanish”,
“Blond  Spanish”?
I(+) =  , I() = .
We have several options:
1. I(Blond) = {X1,X4} - pessimistic
2. I(Blond) = {X1,X4,X2} – optimistic
3. I(Blond) = [{X1,X4}, {X2}] – rough
4. I(Blond) = {(X1,1), (X4,1), (X2,1/2)} assuming that X2 is either Blond or Black and the chances are equal for both colors.
Option 1.
0 = I(Black  Blond) = I(Black)  I(Blond) = {X3}  {X1,X4} = 0
X = I(Black + Blond) = I(Black)  I(Blond) = {X3}  {X1,X4} = {X1,X3,X4}
Option 2.
0 = I(Black  Blond) = I(Black)  I(Blond) = {X2, X3}  {X1,X4,X2} = {X2}
X = I(Black + Blond) = I(Black)  I(Blond) = {X2, X3}  {X1,X4,X2} = X

4. Query Languages (Syntax) – built from
values of attributes (if they are not local, we call them foreign),
functors +, , 
predicates = , <, , …..
Queries form a smallest set T such that:
1. If w  V then w  T.
2. If t1 , t2  T then (t1+t2), (t1t2), (t1)  T
Extended Queries form a smallest set F such that:
1. If t1 , t2  T then (t1 = t2), (t1 < t2), (t1  t2)  F
2. If 1 , 2  F then ~1, ( 1  2), ( 1  2)  F
Query Languages (Local Semantics) –
domain of interpretation has to be established
Example: Query – “blond  Spanish”,
Extended query - “blond  Spanish = black”

18. CERTAIN RULES:
(Headache, Yes)  (Temp, High  Very High)  (Flu, Yes)
(Temp, Normal)  (Flue, No)
POSSIBLE RULES:
(Headache, No)  (Temp, High)  (Flue, Yes), Conf= ½
(Headache, No)  (Temp, High)  (Flue, No), Conf= ½
……………………………………………

19. Dempster-Shafer Theory
based on the idea of placing a number between zero and one to indicate the degree of belief of evidence for a proposition. 19

20. Basic Probability Assignment - function m: 2^X [0,1]
such that: (1) m()=0, (2) [m(Y) : Y X] = 1 /total belief/.
m(Y) – basic probability number of Y.
Belief function over X - function Bel: 2^X  [0,1] such that:
Bel(Y)= [m(Z): Z  Y].
FACT 1: Function Bel: 2^X  [0,1] is a belief function iff
Bel()=0,
(2) Bel(X)= 1,
(3) Bel({A(i): i {1,2,…,n}) =
[(-1)^{|J|+1}Bel({A(i): i  J}) : J  {1,2,…,n}]
for every positive integer n and all subsets A(1), A(2), …, A(n) of X
FACT 2: Basic probability assignment can be computed from:
m(Y) = [ (-1)^{|Y – Z| Bel(Z): Z Y], where Y  X. 20

21. Example: basic probability assignmentd x1 L x2 S x3 P 1 x4 3 R x5 2 x6 x7 H
m_a({x1,x2,x3,x6})=[2+2/3]/7=8/21
m_a({x3,x6,x5})=[1+2/3]/7=5/21
m_a({x3,x6,x4,x7})=[2+2/3]/7=8/21
Basic probability assignment (given)
m({x1,x2,x3,x6})=8/21
m({x3,x6,x5})=5/21
m({x3,x6,x4,x7})=8/21
1) m_a uniquely defined
for x1,x2,x4,x5,x7.
m_a undefined for
x3,x6.
defines attribute m_a
m_a(x1)=m_a(x2) =a1, m_a(x5)=a2,….. 21

22. Example: basic probability assignment
Basic probability assignment – m:
X={x1,x2,x3,x4,x5}
m(x1,x2,x3)=1/2, m(x1,x2)=1/4, m(x2,x4)=1/4
Belief function:
Bel({x1,x2,x3,x5})= ½ + ¼ = ¾, ………..
Focal Element and Core
Y  X is called focal element iff m(Y) > 0.
Core – the union of all focal elements.
Doubt Function Dou: 2^X  [0,1] , Y  X
Dou(Y) = Bel(Y).
Plausibility Function – Pl(Y) = 1 – Dou(Y)
Pl(Y)=[m(Z): Z  Y  ]

23. {1,2}
{1,2}
{1,2}
1/4
{2,3}
3/4
{1,2}
{1,3}
1/2
m({3})=1/2, m({2,3})=1/4,
m({1,2})=1/4.
{1,2}
{1,2}
{1,2}
{3}
1/2
{1}
{2}
Core={1,2,3}
Pl({1,2}) = m({2,3})+m({1,2}) = ½,
Pl({1,3})= m({3})+m({2,3})+m({1,2}) = 1

24. Properties:
- Bel() = Pl() = 0
- Bel(X) = Pl(X) = 1
- Bel(Y)  Pl(Y)
- Bel(Y) + Bel(Y) 1
- Pl(Y) + Pl(Y)  1
- if Y  Z, then Bel(Y)  Bel(Z) and Pl(Y)  Pl(Z)
Bel: 2^X  [0,1] is called a Bayesian Belief Function iff
Bel() = 0
Bel(X) = 1
Bel (Y  Z)= Bel(Y) + Bel(Z), where Y, Z  X, Y  Z = 
Fact: Any Bayesian belief function is a belief function.

25. The following conditions are equivalent:
Bel is Bayesian
All focal elements of Bel are singletons
Bel = Pl
Bel(Y) + Bel(Y) = 1 for all Y  X

26. Dempster’s Rule of Combination
Bel1, Bel2 – belief functions representing two different pieces
of evidence which are independent. Domain = {x1,x2,x3}
Bel1  Bel2 – their orthogonal sum /Dempster’s rule of comb./
m1, m2 – basic probability assignments linked with Bel1, Bel2.
{x1,x2}
1/4
{x1,x2,x3}
3/2
{x2,x4}
{x2}
3/8
3/32
3/16
{x1,x2,x4}
1/16
1/8 m1 m2
(m1 m2)({x1,x2})=3/32+3/16+1/16=11/32
(m1 m2)({x1,x2,x3})=1/8
(m1 m2)({x2})=3/32+3/16+3/32+1/16=7/16
(m1  m2)({x2,x4})=3/32

28. Failing Query Problem

29. Questions?
Thank You