Uncertain Reasoning

Aim:

Sometimes the knowledge in rules is not certain. Rules then may be enhanced by adding information about how certain the conclusions drawn from the rules may be. Our aim in this secion is to describe certainty factors and their manipulation.

Keywords: certainty factor

Plan:

why we might be uncertain measuring uncertainty using certainty factors in rules combining uncertain evidence from more than one source

• Often, experts can't give definite answers.
• May require an inference mechanism that derives conclusions by combining uncertainties.

Certainty Factors

• Logic and rules provide all or nothing answers
• An expert might want to say that something provides evidence for a conclusion, but it is not definite.
• For example, the MYCIN system, an early expert system that diagnosed bacterial blood infections, used rules of this form:

  if   the infection is primary-bacteremia
  and  the site of the culture is one of the sterile sites
  and  the suspected portal of entry is the gastrointestinal tract
  then there is suggestive evidence (0.7) that the infection is bacteroid

• 0.7 is a certainty factor
  

Certainty factors have been quantified using various different systems, including linguistics ones (certain, fairly certain, likely, unlikely, highly unlikely, definitely not) and various numeric scales, such as 0-10, 0-1, and -1 to 1. We shall concentrate on the -1 to 1 version.

Certainty factors may apply both to facts and to rules, or rather to the conclusion(s) of rules.

A "Theory" of Certainty

• Certainty factors range from -1 to +1
• As the certainty factor (CF) approaches 1 the evidence is stronger for a hypothesis.
• As the CF approaches -1 the confidence against the hypothesis gets stronger.
• A CF around 0 indicates that there is little evidence either for or against the hypothesis.

Certainty Factors and Rules

• Premises for rules are formed by the and and or of a number of facts.
• The certainty factors associated with each condition are combined to produce a certainty factor for the whole premise.
• For two conditions P1 and P2:
  CF(P1 and P2) = min(CF(P1), CF(P2))
  CF(P1 or P2) = max(CF(P1), CF(P2))
• The combined CF of the premises is then multiplied by the CF of the rule to get the CF of the conclusion

Example

Assume CF(P1) = 0.6, CF(P2) = 0.4, CF(P3) = 0.2

CF(P1 and P2) = min(0.6, 0.4) = 0.4

CF(0.4, P3) = max(0.4, 0.2) = 0.4

CF(C1) = 0.7 * 0.4 = 0.28

CF(C2) = 0.3 * 0.4 = 0.12

Combining Multiple CF's

• Suppose two rules make conclusions about C.
• How do we combine evidence from two rules?
• Let CF_R1(C) be the current CF for C.
• Let CF_R2(C) be the CF for C resulting from a new rule.
• The new CF is calculated as follows:

  CFR1(C) + CFR2(C) - CFR1(C) * CFR2(C)	when CFR1(C) and CFR2(C) are both positive
  CFR1(C) + CFR2(C) + CFR1(C) * CFR2(C)	when CFR1(C) and CFR2(C) are both negative
  [CFR1(C) + CFR2(C)]/[1 - min(|CFR1(C)|, |CFR2(C)|)]	when CFR1(C) and CFR2(C) are of opposite sign

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  What do certainty factors mean?

  • They are guesses by an expert about the relevance of evidence.
  • They are ad hoc.
  • CF's are tuned by trial and error.
  • CF's hide more knowledge.

  Summary: Uncertain Reasoning

  Certainty factors quantify the confidence that an expert might have in a conclusion that s/he has arrived at. We have given rules for combining certainty factors to obtain estimates of the certainty to be associated with conclusions obtained by using uncertain rules and uncertain evidence.

  Exercises on uncertain reasoning [- try them before clicking ...] Solutions to exercises

  CRICOS Provider Code No. 00098G
  Copyright (C) Bill Wilson, 2002, except where another source is acknowledged. Much of the material on this page is based on an earlier version by Claude Sammut.