1. Dempster/Shaffer Theory of Evidence CIS 479/579 Bruce R. Maxim UM-Dearborn

2. What is it? Means of manipulating degrees of belief that does not require B(A) + B(~A) to be equal to 1 This means that it is possible to believe that something could be both true and false to some degree

3. Example Consider a situation in which you have three competing hypotheses x, y, z There are 8 combinations for true hypotheses {x} {y} {z} {x y} {x z} {y z} {x y z} { }

4. Example Initially you might decide that without any evidence that all three hypotheses are true and assign a weight of 1.0 to the set {x y z} all other sets would be assigned weights of 0.0 With each new pieces of evidence you would begin to decrease the weight assigned to the set {x y z} and increase some of the other weights making sure that the sum of all weights is still 1.0

5. Formally If A is a proposition like the sum of all spots displayed on a pair of 6 sided dice is 7 then set of correct hypotheses would be designated as U The power set of U is made up of all possible subsets of U including both U and the empty set U  P(U)   P(U)

6. Formally We will need to define some function m such that m: P(U)  [0, 1] This function needs to satisfy two conditions m(  ) = 0  m(A) = 1 A  U

7. Formally The function m is called a “basic probability density” function Evidence is regarded as certain if m(F) = 1 So for any A  F m(A) = 0

8. Formally Things become trickier if F is not a singleton set and F  A   Each subset a where m(A)  0 is called a focal element of P(U)

9. Rule of Combination Orthogaonal sum m 1  m 2 If A   [m 1  m 2 ] (A) =  m 1 (X) * m 2 (Y) X  Y = A 1 -  m 1 (X) * m 2 (Y) X  Y = 

10. Rule of Combination If A =  then [m 1  m 2 ] (A) = 0 The function is well defined of the weight of conflict is 1  m 1 (X) * m 2 (Y) = 1 X  Y = 

11. Rule of Combination The denominator of the function 1 -  m 1 (X) * m 2 (Y) X  Y =  is sometimes denoted as 1/k and is used as a normalization factor If 1/k = 0 the then the weight of conflict if 1 and m 1 and m 2 are contradictory and m 1  m 2 is undefined

12. Belief There is also a defined belief function Belief: P(U)  [0, 1] Belief(A) =  m(B) B  A This says that the Belief(A) is the sum of all weights of the subsets formed from A

13. Doubt and Plausibility We can define Doubt(A) = Belief(~A) Plausibility(A) = 1 - Doubt(A) = 1 - Belief(~A)

14. Belief and Plausibility Belief(  ) = 0 Plausibility(  ) = 0 Belief(U) = 1 Plausibility(U) = 1 Plausibility(A) >= Belief(A)

15. Belief and Plausibility Belief(A) + Belief(~A) <= 1 Plausibility(A) + Plausibility(~A) >= 1 If A  B then Belief(A) <= Belief(B) Plausibility(A) <= Plausibility(B)

16. Example S = snow R = rain D = dry U = { S R D} P(U) has 8 elements Assume two pieces of evidence –Temperature is below freezing –Barometric pressure is falling (e.g. storm likely)

17. The following table might be constructed  {S}{R}{D}{S,R}{S,D}{R,D}{S,R,D} Mfreeze00.20.1 0.20.1 0.2 Mstorm00.10.20.10.30.1 Mboth00.282 0.1280.180.051 0.026 The row sums for Mfreeze and Mstorm is 1.0 Mboth is computed from Mfreeze  Mstorm

18. Example Mboth (A) =  Mfreeze(X) * Mstorm(Y) X  Y = A 1 -  Mfreeze(X) * Mstorm(Y) X  Y = 

19. Example Using our table Belief({S R}) =  m(B) = Mboth({S R}) + Mboth({S}) + Mboth({R}) = 0.18 + 0.282 + 0.282 = 0.744 Belief({S R D}) is still 1.0 (sum of Mboth row)

20. Example Using our table and Mfreeze Belief({S R}) =  m(B) = Mfreeze({S R}) + Mfreeze({S}) + Mfreeze({R}) = 0.2 + 0.1 + 0.2 = 0.5

21. Example Using our table and only Mstorm Belief({S R}) =  m(B) = Mstorm({S R}) + Mstorm({S}) + Mstorm({R}) = 0.3 + 0.1 + 0.2 = 0.6

22. Example Our belief based on the combined evidence was stronger than either belief computed from a single source of evidence Note also that Mboth causes larger belief gains from {S} and {R} than for {D}

23. Example If A = {S R} then Doubt(A) = Mboth({D}) = 0.128 Plausibility(A) = 1 – Doubt(A) = 1 – 0.128 = 0.872