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

conditionsm() = 0 m(A) = 1

AU

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 isused 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 functionBelief: P(U) [0 , 1]

Belief(A) = m(B) BA

• 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)

The following table might be constructed

  {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 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