Uncertain Reasoning

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Uncertain Reasoning

• Dempster-Shafer theory

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Dempster-Shafer theory of evidence

• Initial results by Arthur Dempster in the late
  60's.
• Organised and completed by Glenn Shafer in the
  mid 70's.
• Developed by many authors since then.
• Longstanding debate about the philosophical
  foundations and formal semantics of this theory.

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Dempster-Shafer theory of evidence

• Finite domain O - frame of discernment.
• Elementary proposition singleton subset of O,
  i.e. ?.
• Proposition arbitrary subset of O.
• Belief numeric (real) value associated to a
  proposition, Bel(A).

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Dempster-Shafer theory of evidence

• For any A, A1, ..., An c O
• 0 Bel(A) 1.
• Bel( ) 0.
• Bel(O) 1.
• Bel(A1 U ... U An) Si1n Bel(Ai) -
  Si,j1iltjn Bel(Ai n Ai) ...
  (-1)n1 Bel(A1 n ... n An).

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Dempster-Shafer theory of evidence

• Example
• Bel(A1 U A2) Bel(A1) Bel(A2) - Bel(A1 n A2).

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Dempster-Shafer theory of evidence

• Towards a more operational characterisation
• Basic belief mass mapping m from 2O to 0,1.
• m( ) 0.
• SAAcO m(A) 1.
• Bel(B) SAAcB m(A).

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Dempster-Shafer theory of evidence

• Belief
• Bel(B) SAAcB m(A).
• Plausibility
• Pl(B) 1 Bel(O\B) SAAnB? m(A).

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Dempster-Shafer theory of evidence

• From a strict formal standpoint, m, Bel and Pl
  are somehow redundant given m, we have Bel and
  Pl given Bel, we have m (this is a little tricky
  to obtain, but it is true) and Pl and given Pl,
  we have Bel and therefore m. conceptually,
  however, they characterise three different facets
  of beliefs.

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Dempster-Shafer theory of evidence

• m(A) how much an agent believes in proposition A
  alone, with no further assumption about any
  strict subset of A.
• Bel(A) how much an agent believes in proposition
  A, taking into account the beliefs in all subsets
  of A.
• Pl(A) the maximal amount of belief that can be
  attributed to A by an agent.

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Dempster-Shafer theory of evidence

• Interesting property of Bel recall thatBel(A1 U
  A2) Bel(A1) Bel(A2) - Bel(A1 n A2).Assuming
  that A2 O\A1, this expression can be rewritten
  asBel(O) Bel(A1) Bel(A2) Bel(
  ).Therefore,Bel(A1) Bel(O\A1) 1.

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Dempster-Shafer theory of evidence

• This last property is the distinctive feature of
  the Dempster-Shafer theory of evidence this
  theory can explicitly represent ignorance,
  something that is not possible in conventional
  probability theory.