Uncertain Reasoning

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Uncertain Reasoning

• Uncertain reasoning II

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Uncertain reasoning probability

• Let O be a finite universe of discourse. We
  denote the pair (O, 2O) a sample space. Recall
  that we denote the subsets of O events. A
  probability function is any function P 2O ? 0,
  1 that satisfies the following axioms, known as
  the Kolmogorov axioms

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Uncertain reasoning probability

• Kolmogorov axioms
• P(A) 0 for all A c O.
• P(O) 1.
• For any A, B c O, if A n B then P(A U B)
  P(A) P(B).

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Uncertain reasoning probability

• From the Kolmogorov axioms many properties can be
  derived. Some of these properties are
• P( ) 0.
• P(A) 1 for all A c O.
• P(O\A) 1 P(A).
• If A c B then P(A) P(B).
• P(A U B) P(A) P(B) for any A, B c O.
• P(A U B) P(A) P(B) P(A n B).
• ...and many others.

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Uncertain reasoning probability

• Let (O, 2O) be a sample space, P a probability
  function, E c O an arbitrary event such that
  P(E) gt 0. Then, for any event A c O we define
  the conditional probability P(AE) asP(AE)
  P(A n E) / P(E).

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Uncertain reasoning probability

• Chain rule let A1, ..., An c O be events such
  that P(A1 n ... n An-1) gt 0. Then we have
  thatP(A1 n ... n An) P(A1)P(A2A1)P(A3A1n
  A2) ... P(AnA1 n ... n An-1)

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Uncertain reasoning probability

• Total probability rule let H1, ..., Hn be a
  partition of O such that P(Hi) gt 0 for all i 1,
  ..., n. Then we have thatP(A) Si1n
  P(H1)P(AHi)

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Uncertain reasoning probability

• Independence two events A and B are independent
  if and only if P(A n B) P(A)P(B).In a more
  usual (and useful) form, A and B are independent
  given C if and only ifP(A n B C)
  P(AC)P(BC).

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Uncertain reasoning probability

• Let O be a universe and T be a refinement of O.
  Let P be a probability function defined on T.
  Then, there is a probability function P' defined
  on O, such thatP'(?) S? ? ?(?) P(?).

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Uncertain reasoning probability

• Let U, V be general universes such that U is a
  projection of V. Let P be a probability function
  defined on V. Then, there is a probability
  function P' defined on U, such thatP'(u) Sv
  ? ?(u) P(v).