The document discusses uncertainty in expert systems, highlighting the concepts of certainty factors (CF) as a method to evaluate and manage uncertainty compared to Bayesian reasoning. It details how CFs can be computed from evidence to support or refute hypotheses, including methods for combining multiple rules and the interpretation of CF values. Overall, the CF framework assists in improving decision-making by providing a quantifiable measure of belief and disbelief in the conclusions drawn from uncertain data.

1. Uncertainty in Expert
Systems : Certainty
Factor
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2. Uncertainty
• Uncertainty is the lack of exact knowledge that
would enable us to reach a fully reliable solution.
o Classical logic assumes perfect knowledge exists:
IF A is true
THEN B is true
• Describing uncertainty:
o If the antecedence A is true, then consequent B is
true to a certain probability
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3. Sources of uncertainty
• Weak implications: Want to be able to capture
associations and correlations, not just cause and
effect.
• Imprecise language:
o How often is “sometimes”?
o Can we quantify “often,” “sometimes,” “always?”
• Unknown data: In real problems, data is often
incomplete or missing.
• Differing experts: Experts often disagree, or have
different reasons for agreeing.
o Solution: attach weight to each expert 3

4. Two approaches
• Bayesian reasoning – Bayesian rule
- (Probability theory)
• Certainty factors
- Difference between belief and disbelief.
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5. Certainty factors
• Certainty factor (CF) is an alternative to
Bayesian method of dealing with uncertainty.
This method was originally developed for the
MYCIN system.
• CF was originally defined as the difference
between belief and disbelief.
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6. • In formula form:
where
o CF is the certainty factor in the hypothesis H due to
evidence E
o MB is the measure of increased belief in H due to E
o MD is the measure of increased disbelief in H due
to E
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7. • p(H) is the prior probability of hypothesis H being true;
• p(H|E) is the probability that hypothesis H is true given
evidence E.
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8. • According to CF definition, we can deduce some
of its characteristics
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9. Interpreting uncertainty
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10. • A positive CF means the evidence supports the hypothesis
since MB > MD.
• A CF = 1 means that the evidence definitely proves the
hypothesis.
• A negative CF means that there is more reason to disbelief a
hypothesis than to belief it.
example, a CF = - 0.7 means that the disbelief is 70%
greater than the belief.
• A CF = 0 means one of two possibilities. First, a CF = MB - MD
= 0 could mean that both MB and MD are 0, second both are
having same nonzero value, where in either cases, the belief is
canceled out by the disbelief.
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11. • CF is a way of combining belief and disbelief into
a single number.
• The certainty factor can be used to rank
hypothesis in order of importance.
• For example, If a patient has certain symptoms
which suggest several possible diseases, then
the disease with the highest CF would be the
one that is first investigated by ordering tests.
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12. • The certainty factor assigned by a rule is propagated
through the reasoning chain.
• This involves establishing the net certainty of the rule
consequent when the evidence in the rule antecedent is
uncertain:
cf (H,E) = cf (E) x cf
For example,
IF sky is clear
THEN the forecast is sunny {cf 0.8}
given the current certainty factor of sky is clear is 0.5,
then cf (H,E) = 0.5 × 0.8 = 0.4
This result can be interpreted as “It may be sunny”.
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13. Combining Certainty Factors
• For conjunctive rules such as
the certainty of hypothesis H, is established as follows:
𝑐𝑓 𝐻, 𝐸1 ∩ 𝐸2 ∩ 𝐸3 … ∩ 𝐸𝑛 = min [cf (E1), cf (E2),..., cf (En)] x cf
For example,
IF sky is clear
AND the forecast is sunny
THEN the action is ‘wear sunglasses’ {cf 0.8}
and the certainty of sky is clear is 0.9 and the certainty of the
forecast of sunny is 0.7, then
𝑐𝑓(𝐻, 𝐸1 ∩ 𝐸2) = min [0.9, 0.7] × 0.8 = 0.7 × 0.8 = 0.56
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14. • For disjunctive rules such as
the certainty of hypothesis H, is established as follows:
𝑐𝑓 𝐻, 𝐸1 ∪ 𝐸2 ∪ 𝐸3 … ∪ 𝐸𝑛 = max [cf (E1), cf (E2),..., cf (En)] x cf
For example,
IF sky is overcast
OR the forecast is rain
THEN the action is ‘take an umbrella’ {cf 0.9}
and the certainty of sky is overcast is 0.6 and the certainty of
the forecast of rain is 0.8, then
𝑐𝑓(𝐻, 𝐸1 ∪ 𝐸2) = = max [0.6, 0.8] × 0.9 = 0.8 × 0.9 = 0.72
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15. • Multiple Rules with Same Conclusion
Suppose we have two rules R1 and R2 with the same conclusion
Z
Rule 1: IF A is X
THEN C is Z {cf 0.8}
Rule 2: IF B is Y
THEN C is Z {cf 0.6}
What certainty should be assigned to object C having value Z
if both Rule 1 and Rule 2 are fired?
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16. • Multiple Rules with Same Conclusion
To calculate a combined certainty factor we can use the following
equation:
where:
cf1 is the confidence in hypothesis H established by Rule 1;
cf2 is the confidence in hypothesis H established by Rule 2;
|cf1| and |cf2| are absolute magnitudes of cf1 and cf2,
respectively.
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17. Example 1
Consider the previous rules:
Rule 1: IF A is X
THEN C is Z {cf 0.8}
Rule 2: IF B is Y
THEN C is Z {cf 0.6}
if we assume that
𝑐𝑓 𝐸1 = 𝑐𝑓 𝐸2 = 1
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18. Consider the previous rules:
𝑐𝑓1 𝐻, 𝐸1 = 𝑐𝑓 𝐸1 × 𝑐𝑓 = 1.0 × 0.8 = 0.8
𝑐𝑓2 𝐻, 𝐸2 = 𝑐𝑓 𝐸2 × 𝑐𝑓 = 1.0 × 0.6 = 0.6
From the equation we obtain
𝑐𝑓 𝑐𝑓1, 𝑐𝑓2 = 𝑐𝑓1 𝐻, 𝐸1 + 𝑐𝑓2 𝐻, 𝐸2 − [1 − 𝑐𝑓1 𝐻, 𝐸1 ]
= 0.92
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19. Again Consider the previous rules:
Rule 1: IF A is X
THEN C is Z {cf 0.8}
Rule 2: IF B is Y
THEN C is Z {cf 0.6}
Now, if we assume that
𝑐𝑓 𝐸1 = 1, 𝑐𝑓 𝐸2 = −1
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20. Consider the previous rules:
𝑐𝑓1 𝐻, 𝐸1 = 𝑐𝑓 𝐸1 × 𝑐𝑓 = 1.0 × 0.8 = 0.8
𝑐𝑓2 𝐻, 𝐸2 = 𝑐𝑓 𝐸2 × 𝑐𝑓 = −1.0 × 0.6 = −0.6
From the equation we obtain
𝑐𝑓 𝑐𝑓1, 𝑐𝑓2 =
𝑐𝑓1 𝐻, 𝐸1 + 𝑐𝑓2 𝐻, 𝐸2
1 − min[ 𝑐𝑓1 𝐻, 𝐸1 , 𝑐𝑓1 𝐻, 𝐸1 ]
= 0.5
The combined certainty factor now drop because one
evidence confirms a hypothesis but another discounts it.
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21. Example 2
• We have the following rules
R1: if a and b then x (cf = 0.5)
R2: if c or d then x (cf = 0.7)
• We have the following Input:
a, with certainty 1.0
b, with certainty 0.8
c, with certainty 0.9
d, with certainty 0.7
Then, Compute CF values for x:
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22. • CF(a and b) = MIN{1.0, 0.8} = 0.8
==> CF1(x) = 0.8 * 0.5 = 0.4
• CF(c or d) = MAX{0.9, 0.7} = 0.9
==> CF2(x) = 0.9 * 0.7 = 0.63
• CF(CF1, CF2) = 0.4 + 0.63 - 0.4*0.63 = 0.778
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23. Example 3
• Suppose we have the following rule R1:
if (P1 and P2 and P3) or (P4 and not P5 then C1 (0.7)
and C2 (-0.5)
• Given the certainty factors of P1, P2, P3, P4, P5 are as follows:
CF(P1) = 0.8,
CF(P2) = 0.7,
CF(P3) = 0.6,
CF(P4) = 0.9,
CF(P5) = -0.5,
• What are the certainty factors associated with conclusions C1
and C2 after using rule R1?
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24. • Solution:
For P1 and P2 and P3, the CF is
min(CF(P1), CF(P2), CF(P3)) = min(0.8, 0.7, 0.6) = 0.6.
Call this CFA.
For not P5, the CF is -CF(P5) = 0.5.
For P4 and not P5, the CF is min(0.9, 0.5) = 0.5.
Call this CFB.
For (P1 and P2 and P3) or (P4 and not P5), the CF is:
max(CFA, CFB) = max(0.6, 0.5) = 0.6.
Thus
CF(C1) = 0.7 * 0.6 = 0.42
CF(C2) = -0.5 * 0.6 = -0.30
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