Dempster/Shaffer Theory of Evidence

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Dempster/ShafferTheory of Evidence

• CIS 479/579
• Bruce R. Maxim
• UM-Dearborn

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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

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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

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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

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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)

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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

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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

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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)

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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 ?

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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 ?

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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

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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

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Doubt and Plausibility

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

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Belief and Plausibility

• Belief(?) 0
• Plausibility(?) 0
• Belief(U) 1
• Plausibility(U) 1
• Plausibility(A) gt Belief(A)

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Belief and Plausibility

• Belief(A) Belief(A) lt 1
• Plausibility(A) Plausibility(A) gt 1
• If A ? B then
• Belief(A) lt Belief(B)
• Plausibility(A) lt Plausibility(B)

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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)

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The following table might be constructed
The row sums for Mfreeze and Mstorm is 1.0 Mboth
is computed from Mfreeze ? Mstorm
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Example

• Mboth (A) ? Mfreeze(X) Mstorm(Y)
• X ? Y A
• 1 - ? Mfreeze(X) Mstorm(Y)
• X ? Y ?

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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)

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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

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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

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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

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Example

• If A S R then
• Doubt(A) Mboth(D) 0.128
• Plausibility(A) 1 Doubt(A)
• 1 0.128
• 0.872