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

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Examples – Datasets (Information Systems)

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Information System Color_of_HairNationality X1Blond X2 Spanish X3BlackSpanish X4BlondGerman How to interpret term “blond”? How to interpret “blond + Spanish”, “Blond Spanish”? 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. I(+) = , I( ) = . 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

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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”

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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 ……………………………………………

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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.

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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.

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Example: basic probability assignment Xabcd x1x1 0L x2x2 0SL x3x3 P1L x4x4 3R1L x5x5 22L x6x6 P2L x7x7 3P2H 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 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,…..

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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 ]

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{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}

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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 1)Bel( ) = 0 2)Bel(X) = 1 3)Bel (Y Z)= Bel(Y) + Bel(Z), where Y, Z X, Y Z = Fact: Any Bayesian belief function is a belief function.

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The following conditions are equivalent: 1)Bel is Bayesian 2)All focal elements of Bel are singletons 3)Bel = Pl 4)Bel(Y) + Bel( Y) = 1 for all Y X

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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

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Failing Query Problem

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Thank You Questions?

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