UNCERTAINTY

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UNCERTAINTY

• Introduction to Artificial Intelligence

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UNCERTAINTY

• Much of the information we deal with is inexact,
  situations such as "most birds can fly", "babies
  cry a lot", etc.
• We have information about what is likely.
• When using non-monotonic and default reasoning,
  we record what is most likely, but then reason
  about it as if it were true.

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UNCERTAINTY

• Work in the area of uncertainty tries to find
  ways to measure the certainty of something, and
  use the measure of uncertainty in the reasoning
  process.

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Sources of Uncertainty

• With respect to facts
• 1. Incomplete Information
• - May not know everything.
• Medicine, do not want to run every test
• - May not want to wait
• Patient could die
• Need to use what information is available

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Sources of Uncertainty

• 2. Vagueness or Ambiguity in the predicate -
• does this predicate apply to the situation?
• Old people need glasses, what qualifies as old?
  Should this be applied to our given situation).
• 3. Irreducible Uncertainty
• is the person sincere?

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Sources of Uncertainty

• With respect to rules
• Rules are usually heuristics that experts use in
  given situations.
• Not necessarily perfect, or exactly correct.

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Requirements for an Approach to Uncertainty

• 1. Uncertainty as to the applicability of the
  conditions of rules.
• example
• ((P Q) (R v S)) --gt T
• if we are completely certain about P, Q, R and
  S, it is easy to conclude T.

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Requirements for an Approach to Uncertainty

• What if we are certain of P, think Q is possible,
  R is probably and S high unlikely?
• Can we conclude T?
• Can we conclude T with some reduced level of
  confidence?

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Requirements for an Approach to Uncertainty

• 2. Way to propagate uncertainty attaching to
  rules
• We may be uncertain about the rule.
• P -gt Q
• even if we are certain about P, we may be
  uncertain about Q because we are not completely
  certain about the rule.

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Requirements for an Approach to Uncertainty

• 3. Combine evidence from several rules
• If several rules point to the same conclusion, we
  would like to be able to show greater certainty
  about the conclusion than if only one rule
  pointed to that conclusion.

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Requirements for an Approach to Uncertainty

• 5 questions to consider
• How is confidence in evidence measured?
• How is confidence in rules measured?
• How is evidence combined?
• Given uncertain evidence and an uncertain rule
  what confidence can be placed in the conclusion?
• Given the same uncertain conclusion from several
  rules what overall confidence should be placed
  in that conclusion?

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Some Approaches to Uncertainty

• Classical Probability Theory
• Bayes' Theorem
• Certainty Factors
• Fuzzy Logic
• Qualitative

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Classical Probability Theory

• likelihood is expressed by a value of 0 to 1.
• Conjunction (p and q) (p q)
• Disjunction (p or q) (p q - 2pq)
• negation p (1 - p)
• Application
• Good for first and third questions.
• Second and fourth cause problems

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Bayes' Theorem

• Bayes' theorem - used for conditional probability
• P(disease symptom) - probability of disease
  given some symptom.
• According to Bayes' theorem
• P (disease symptom) / P (symptom)

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Bayes' Theorem

• Helps with combining evidence
• However, for the first two considerations (How is
  confidence in evidence measured? How is
  confidence in rules measured?) problems are
• number of probabilities required
• experts have a difficult time providing such
  probabilities.

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Certainty Factors

• Used in MYCIN based on the Depmster - Shafer
  theory of evidence.
• Uses measures of belief (MB) and measure of
  disbelief (MD)
• MB and MD are each assigned a value between 0 and
  1, but are not required to sum to 1.

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Certainty Factors

• As evidence accumulated, MB and MD change,
  causing changes in CF
• MYCIN put cf in rules between 0 and 1 (lt0 means
  negative confidence, and we would not include
  such rules in the knowledge base).

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Certainty Factors

• Conjunction of two propositions is the minimum
  of the CF's of the two propositions.
• Disjunction of two propositions is the maximum
  of the CF's of the two propositions.
• Negation -1 CF

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Certainty Factors

• Combining CF for a set of conditions with the
  rule strength to get a CF for the conclusion
• multiply the CF of the conditions by the rule
  strength

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Certainty Factors

• Combining evidence from several rules
• sum independent evidence and subtract the product

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Certainty Factors

• Certainty factors provide a method to handle all
  five questions
• However is not justifiable in the way that
  classical probability theory or Bayes' Theorem is
  justifiable.

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Fuzzy Logic

• Based on fuzzy set theory of Zadeh.

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Membership Function A Positive Integer Much Less
Than 10
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Non-fuzzy Set A Positive Integer Less Than 5
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Non-Fuzzy Set

• Vertical lines would be better

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Fuzzy Sets Representing Negation
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Fuzzy Logic

• How to represent negation.
• Member(complement of p)
• 1 - member (p)

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Fuzzy Logic

• The problem with fuzzy logic is that we can't
  really be sure what the precision represents.
• Truth values have infinite precision, but can we
  really express "truth" with this level of
  precision?
• Not likely.

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Fuzzy Logic

• However, it is computationally tractable and some
  applications have some good results.
• There is a fair amount of work being done in this
  area, and may prove more useful as things
  advance.

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Qualitative

• Some people don't like the idea of attaching
  numbers of any kind to a reasoning process.
• They want to use possibly or probably to express
  these concepts.
• Not computationaly tractable, so even if used in
  the user interface, they tend to be converted to
  numbers for inference purposes.

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Summary

• We need ways to reason about inexact things.
• Some formal techniques exist (probability theory
  and fuzzy logic)
• Formal techniques tend only to be useful in
  certain domains

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Summary

• More "ad hoc" techniques (like certainty factors)
  have been developed.
• These techniques are not justifiable in the way
  formal techniques are they have been useful and
  can be programmed (a key feature)

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Evaluation of Uncertainty Approaches