The Combination of Evidence in DempsterShafer Theory

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The Combination of Evidence in Dempster-Shafer
Theory

• Kari Sentz
• Ph.D. Student in Systems Science
• Binghamton University
• Epistemic Uncertainty Project
• Sandia National Laboratories

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Dempster-Shafer Theory (DST)

• A mathematical theory of evidence
• Probabilities on a set which is related to a set
  of situations
• Originally defined in finite space, many recent
  efforts have taken this to the continuous domain.
• Important elements of the theory
• Belief and Plausibility measures
• basic probability assignment (m)
• Rules of Combination

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Basic Probability Assignment

• m P(X) ? 0,1
• m(Ø) 0
• m(A) expresses the proportion to which all
  available and relevant evidence supports the
  claim that a particular element of X belongs to A
• pertains only to set A
• does not imply any additional claims regarding
  the subsets of A
• any additional evidence of the subsets of A, i.e.
  B ? A, the claim must be expressed by another
  value m(B)

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Belief, Plausibility, and Basic Probability
Assignment

• For all sets A?P(X)
• where A-B is the cardinality of the set
  difference of A and B

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Belief and Plausibility

• Transformation
• Relation Pl(A) ? Bel (A)
• Interval Bel (A), Pl (A)

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Set Theoretic Classification for Rules of
Combination

• Conjunctive operation (AND, ?)
• corresponds to the situation where all sources
  are considered reliable
• Disjunctive operation (OR, ?)
• corresponds to the situation where there is only
  one reliable source among many
• Trade-off operation
• corresponds to any situation that is not clearly
  one of the above

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Important Primitive Calculations to
Dempster-Type Rules
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Dempster-Type Rules
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Averaging

• Weighted Averaging or Mixing
• Generalized average for probability distributions
• Discount and Combine
• Convolutive X-Averaging
• Generalized average for scalar numbers

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Simple Example
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Relevant Calculations
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Inagakis Rule for Simple Example
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Disjunctive Consensus Rule for Simple Example
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Important Points in Conflict Assessment
CONTEXT
COMBINATION OPERATION Type of Operation Algebraic
Properties Handling of conflict Advantages Disadva
ntages
EVIDENCE Type of evidence Amount Accuracy
SOURCES Type Number Reliability Dependency Conflic
t
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Conclusions

• Many of the criticisms of DST have rejected the
  theory on the basis of the Dempster rule of
  combination.
• There are many ways to combine evidence in DST.
• The critical concern for the selection of a
  combination operation is the nature of conflict
  and how it is handled by a particular combination
  rule.