Uncertainity

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Uncertainity

• AI Lecture 9 by Zahid Anwar

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Is abduction necessary?

• Although abduction is unsound it is often
  essential to solving problems
• The correct version of battery rule is not
  particularly useful in diagnosing car troubles
  since its premise bad-battery is our goal and its
  conclusions are the observable symptoms we must
  work with
• Modus Ponens can not be applied and the rule must
  be used in an abductive fasion

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Abduction

• This is generally true of diagnostic and other
  expert systems
• Fault or diseases cause symptoms, but diagnosis
  must work from the symptoms back to the cause
• Uncertainty results from the use of abductive
  inference as well as from attempts to reason with
  missing or unreliable data
• To get around this problem, we can attach some
  measure of confidence to the conclusion

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Example

• For example, although battery failure does not
  always accompany the failure of a cars lights
  and starters, it almost does, and confidence in
  this rule is justifiably high.
• There are several ways of dealing with
  uncertainty that results from heuristic rules
• Bayesian Approach
• Stanford Certainty theory
• Zadehs fuzzy set theory
• Non-monotonic reasoning

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Bayesian Probability Theory

• The Bayesian approach to uncertainty is based on
  formal probability theory
• Assuming random distribution of events,
  probability theory allows the calculation of more
  complex probabilities from previously known
  results
• In mathematical theory of probability individual
  probability instances are worked out by sampling
  and combinations of probabilities are worked out
  using the rule such as

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Bayesian Probability Theory

• Probability (A and B) probability (A )
  probability ( B ) given that A and B are
  independent results
• One of the most important results of probability
  theory is Bayes theorem
• Bayes results provide a way of computing the
  probability of a hypothesis following from a
  particular piece of evidence , given only the
  probabilities with which the evidence follows
  from actual cases

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Bayess theorem states

• P (H E) P (E H) P (H)
• S (from k1 to n) ( P(EHK) P
  (HK))
• Where P(HE) is the probability that H is true
  given evidence E
• P (H) is the probability that H is true overall
• P(EH) is the probability of observing evidence E
  when H is true
• N is the number of possible hypothesis

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Semantics of Predicate Calculus

• The truth of expressions depends on the mapping
  of constants, variables, predicates and functions
  into objects and relations of the domain of
  discourse
• The truth of relationships in the domain
  determines he truth of corresponding expressions
• Friends(george, susie)
• Friends(george, Kate)

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Interpretation

• An Interpretation is an assignment of the D to
  each of the constants, variables, predicates and
  functions

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Satisfy

• An Interpretation that makes a sentence true is
  said to satisfy that sentence
• An Interpretation that satisfies every member of
  a set of expressions is said to satisfy the set

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Logically Follows

• An expression X, logically follows from a set of
  predicate calculus expressions S if every
  Interpretation that satisfies S also satisfies X
• The function of a logical inference is to produce
  new sentences that logically follows a given set
  of expressions

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Sound

• When every sentence X produced by an inference
  rule operating on a set S of logical expressions
  logically follows from S, the inference rule is
  said to be sound
• If the inference rule is able to produce every
  sentence that logically follows from S, then it
  is said to be complete
• Modus Ponens and resolution are examples of
  inference rules that are sound

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Example of an Interpretation

• FOR ALL x human (x) ?mortal (x) ?
• Human(socrates) ? dead(socrates)

Every Human Dies