Dempster/Shaffer Theory of Evidence

Dempster/ShafferTheory of Evidence CIS 479/579 Bruce R. Maxim UM-Dearborn

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

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} { }

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

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)

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

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

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)

Rule of Combination • Orthogaonal sum m1 m2 If A   [m1 m2] (A) =  m1(X) * m2(Y) X  Y = A 1 -  m1(X) * m2(Y) X  Y = 

Rule of Combination If A =  then [m1 m2] (A) = 0 • The function is well defined of the weight of conflict is 1  m1(X) * m2(Y) = 1 X  Y = 

Rule of Combination • The denominator of the function 1 -  m1(X) * m2(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 m1 and m2 are contradictory and m1 m2 is undefined

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

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

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

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)

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)

 {S} {R} {D} {S,R} {S,D} {R,D} {S,R,D} Mfreeze 0 0.2 0.1 0.1 0.2 0.1 0.1 0.2 Mstorm 0 0.1 0.2 0.1 0.3 0.1 0.1 0.1 Mboth 0 0.282 0.282 0.128 0.18 0.051 0.051 0.026 The following table might be constructed The row sums for Mfreeze and Mstorm is 1.0 Mboth is computed from Mfreeze  Mstorm

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

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)

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

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

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}

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

Dempster/Shaffer Theory of Evidence