05<Rule-based Uncertain Reasoning >-1

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Lecture 05 Rule-based Uncertain Reasoning

• Topics
• Information Uncertainty
• Bayesian Inference Model
• Certainty Factor Model
• Discussion

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Information uncertainty

• Information can be incomplete, inconsistent,
  uncertain, or imprecise.
• Incomplete missing knowledge
• Inconsistent conflicting Knowledge, such as from
  different experts
• Uncertainty lack of exact knowledge
• Imprecise ambiguous knowledge, such as terms of
  often, sometimes, frequently and hardly ever

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Information uncertainty
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Bayesian inference model

• Representation (Cause-effect rule)
• IF H is true
• THEN E is true with probability p
• Semantics
• IF event H occurs, THEN the probability that
  event E will occur is p, or p(EH) p.
• p(H), p(E) prior probability
• p(E?H), p(H?E) conditional probability
• Inference
• Given p(H) and p(EH)
• Find p(HE).

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Bayesian inference model

• Bayesian rule

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Bayesian inference model

• Single evidence and single hypothesis
• Single evidence and multiple hypotheses

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Bayesian inference model

• Multiple evidence and multiple hypotheses
• Multiple evidence and multiple hypotheses under
  conditional independence

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Bayesian inference model

• Example Naïve Bayesian Classifier

P(Hi)
H1
H2
H3
Hm
P(EjHi)
E1
E3
E2
En
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Characteristics of Bayesian inference

• Humans are hard eliciting probability values
  consistent with the Bayesian rules.
• Humans tend to make different assumptions when
  assessing the conditional and prior
  probabilities.
• Bayesian Inference is most appropriate in the
  domains where reliable statistical data exist,
  for instance, forecasting.
• Bayesian inference is of exponential complexity,
  and thus is impractical for large knowledge
  bases.

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Certainty factor model

• Representation (diagnostic rule)
• IF ?evidence E?
• THEN ?hypothesis H? cf
• Semantics
• Given that evidence E has occurred, we have cf
  degree of belief that hypothesis H will happen.
• 1 lt cf lt 1
• Inference
• Given cf(E) and cf
• Find cf(H, E)

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Certainty factor model

• Meaning of certainty factors

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Certainty factor model

• Single evidence and single hypothesis
• IF ?evidence E?
• THEN ?hypothesis H? cf
• cf (H, E) cf (E) ? cf
• Conjunctive rules
• cf (H, E1?E2?...?En) min cf (E1), cf (E2),...,
  cf (En) ? cf

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Certainty factor model

• Disjunctive rules
• cf (H, E1?E2?...?En) max cf (E1), cf (E2),...,
  cf (En) ? cf

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Certainty factor model

• Multiple rules conclude with the same hypothesis
• cfi is the confidence in hypothesis H established
  by Rule i

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Characteristics of Certain factors

• Certainty factors are used in domains where the
  probabilities are not known or are too difficult
  or expensive to obtain, for instance in medicine.
  
• The evidential reasoning mechanism can manage
  incrementally acquired evidence, the conjunction
  and disjunction of hypotheses, as well as
  evidence with different degrees of belief.

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Discussion

• More inexact reasoning models
• Likelihood inference
• Dempster-Shafer evidential reasoning
• Fuzzy inference