Fundamentals of Dempster-Shafer Theory

presented by

Zbigniew W. RasUniversity of North Carolina, Charlotte, NC

Warsaw University of Technology, Warsaw, Poland

College of Computing and InformaticsUniversity of North Carolina, Charlotte

www.kdd.uncc.edu

Examples – Datasets (Information Systems)

Information System   Color_of_Hair Nationality

X1 Blond  

X2   Spanish

X3 Black Spanish

X4 Blond German

How to interpret term “blond”?

How to interpret “blond + Spanish”,“Blond Spanish”?

We have several options:1. I(Blond) = {X1,X4} - pessimistic2. I(Blond) = {X1,X4,X2} – optimistic3. I(Blond) = [{X1,X4}, {X2}] – rough4. 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.

I(+) = , I() = .

Option 1. 0 = I(Black Blond) = I(Black) I(Blond) = {X3} {X1,X4} = 0X = 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

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), (t1t2), (t1) T

Extended Queries form a smallest set F such that:1. If t1 , t2 T then (t1 = t2), (t1 < t2), (t1 t2) F2. 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”

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……………………………………………

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.

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(1)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.

Example: basic probability assignment

X a b c d

x1 0 L

x2 0 S L

x3 P 1 L

x4 3 R 1 L

x5 2 2 L

x6 P 2 L

x7 3 P 2 H

m_a({x1,x2,x3,x6})=[2+2/3]/7=8/21m_a({x3,x6,x5})=[1+2/3]/7=5/21m_a({x3,x6,x4,x7})=[2+2/3]/7=8/21

Basic probability assignment (given)m({x1,x2,x3,x6})=8/21m({x3,x6,x5})=5/21m({x3,x6,x4,x7})=8/21

defines attribute m_a

1) m_a uniquely defined for x1,x2,x4,x5,x7.2)m_a undefined for x3,x6.

m_a(x1)=m_a(x2) =a1, m_a(x5)=a2,…..

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 XDou(Y) = Bel(Y).Plausibility Function – Pl(Y) = 1 – Dou(Y)Pl(Y)=[m(Z): Z Y ]

{1,2}{1,2}1/4

{1,2}{2,3}3/4

{1,2}{1,3}1/2

{1,2}{1}0

{1,2}{2}0

{1,2}{3}1/2

m({3})=1/2, m({2,3})=1/4,m({1,2})=1/4.

Pl({1,2}) = m({2,3})+m({1,2}) = ½, Pl({1,3})= m({3})+m({2,3})+m({1,2}) = 1

Core={1,2,3}

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 iff1)Bel() = 02)Bel(X) = 13)Bel (Y Z)= Bel(Y) + Bel(Z), where Y, Z X, Y Z =

Fact: Any Bayesian belief function is a belief function.

The following conditions are equivalent:

1)Bel is Bayesian2)All focal elements of Bel are singletons3)Bel = Pl 4)Bel(Y) + Bel(Y) = 1 for all Y X

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}1/4

{x2}3/8

{x2}3/32

{x2}3/16

{x2}3/32

{x1,x2,x4}3/8

{x1,x2}3/32

{x1,x2}3/16

{x2,x4}3/32

{x1,x2,x3}1/4

{x1,x2}1/16

{x1,x2,x3}1/8

{x2}1/16

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

Failing Query Problem

Thank You

Questions?