Certainty Factor
Model
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Introduction
Probability theory has been called by mathematicians a theory of
reproducible uncertainty
Besides the subject probability theory, alternative theories were
specifically developed to deal with human belief rather than the
classic frequency interpretation of probability
All these theories are examples of inexact reasoning
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Uncertainty and rules
Sources of Uncertainty in rules
The goal of the knowledge engineer is to minimize or eliminate these
uncertainties
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Sources of Uncertainty in rules
Besides the possible errors involved in the creation of rules, there are
uncertainties associated with the assignment of likelihood values.
For probabilistic reasoning, these uncertainties are with the sufficiency, LS,
and necessity, LN, values
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Sources of Uncertainty in rules
Uncertainty with the likelihood of the consequent.
For probabilistic reasoning, written as
P(H I E) for certain evidence and
P(H I e) for uncertain evidence
Another source of uncertainty is the combining of the evidence.
Should the evidence be combined. as in the following?
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Certainty factor
Another method of dealing with uncertainty uses certainty factors
Difficulties with the Bayesian Method
Bayes Theorems accurate use depends on knowing many
probabilities
For example, to determine the probability of a specific disease,
given certain symptoms as:
where the sum over j extends to all diseases, and:
Di is the i'th disease,
E is the evidence,
P(Di) is the prior probability of the patient having the Diseasei
before any evidence is known
P(E I Di) is the conditional probability that the patient will exhibit
evidence E, given that disease Di is present
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Difficulties with the Bayesian
Method
A convenient form of Bayes Theorem that expresses the
accumulation of incremental evidence like this is
where E2 is the new evidence added to yield the new
augmented evidence:
Although this formula is exact, all these probabilities are not
generally known
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Belief and disbelief
Another major problem was the relationship of belief and disbelief
the theory of probability states that:
P(H) + P(H') = 1
and so:
P(H) = 1 - P(H')
For the case of a posterior hypothesis that relies on evidence, E:
(1) P(H | E) = 1 - P(H' | E)
Experts were extremely reluctant to state their knowledge in the form of
equation (1).
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Belief and disbelief
For example, consider a MYCIN rule:
IF 1) The stain of the organism is gram positive,
and
2) The morphology of the organism is coccus,
and
3) The growth conformation of the organism is chains
THEN There is suggestive evidence (0.7) that the identity of the
organism is streptococcus
in terms of posterior probability as:
where the Ei correspond to the three patterns of the antecedent
An expert would agree to equation (2), they became uneasy and
refused to agree with the probabilistic result:
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Belief and disbelief
The fundamental problem is that
while P(H | E) implies a cause-and-effect relationship between E and H
there may be no cause-and-effect relationship between E and H'
Yet the equation:
P(H | E) = 1 P(H' | E)
implies a cause-and-effect relationship between E and H' if there is
a causeand-effect between E and H
Certainty factors representing uncertainty
ordinary probability
associated with the frequency of reproducible events
epistemic probability or the degree of confirmation
it confirms a hypothesis based on some evidence (another example
of the degree of likelihood of a belief. )
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Measures of belief and disbelief
In MYCIN, the degree of confirmation was originally defined as the
certainty factor
the difference between belief and disbelief:
CF(H,E) = MB(H,E) - MD(H,E)
where
CF is the certainty factor in the hypothesis H due to evidence E
MB is the measure of increased belief in H due to E
MD is the measure of increased disbelief in H due to E
Certainty factor
is a way of combining belief and disbelief into a single number
can be used to rank hypotheses in order of importance
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Measures of belief and disbelief
The measures of belief and disbelief were defined in terms of
probabilities by:
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Measures of belief and disbelief
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Measures of belief and disbelief
The certainty factor, CF, indicates the net belief in a hypothesis
based on some evidence.
positive CF means the evidence supports the hypothesis since MB >
MD
CF = 1 means that the evidence definitely proves the hypothesis.
CF = 0 means one of two possibilities
1.
2.
negative CF means that the evidence favors the negation of the
hypothesis since MB < MD
CF = MB - MD = 0 could mean that both MB and MD are 0
that is, there is no evidence
The second possibility is that MB = MD and both are nonzero
the belief is cancelled out by the disbelief
there is more reason to disbelieve a hypothesis than to believe it
With certainty factors there are no constraints on the individual
values of MB and MD. Only the difference is important
CF = 0.70 = 0.70 - 0
= 0.80 - 0.10
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Measures of belief and disbelief
Certainty factors allow an expert to express a belief without
committing a value to the disbelief
CF(H,E) + CF(H',E) = 0
means that if evidence confirms a hypothesis by some value CF(H | E)
the confirmation of the negation of the hypothesis is not 1 - CF(H | E)
which would be expected under probability theory
CF(H,E) + CF(H',E) 1
For the example of the student graduating if an "A" is given in the
course.
CF(H,E) = 0.70
CF (H, E) = - 0.70
which means:
(6) I am 70 percent certain that I will graduate if I get an 'A' in this course
(7) I am -70 percent certain that I will not graduate if I get an 'A' in this
course
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Measures of belief and disbelief
Certainty factors are defined on the interval:
where
0 means no evidence.
values greater than 0 favor the hypothesis
less than 0 favor the negation of the hypothesis
The above CF values might be elicited by asking:
How much do you believe that getting an A will help you graduate?
If the evidence is to confirm the hypothesis or:
How much do you disbelieve that getting an A will help you graduate?
An answer of 70 percent to each question will set
CF(H | E) = 0.70 and
CF (H' | E) = -0.70.
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Calculating with certainty factors
There were difficulties with the definition CF=MB-MD
E.g., 10 pieces of evidence might produce a MB = 0.999 and one
disconfirming piece with MD = 0.799 could then give:
CF = 0.999 - 0.799 = 0.200
In MYCIN, CF > 0.2 for the antecedent to activate the rule
Threshold value is an ad hoc way of minimizing the activation of
rules which only weakly suggest a hypothesis
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Calculating with certainty factors
The definition of CF was changed in MYCIN in 1977 to be:
To soften the effects of a single piece of disconfirming evidence
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Calculating with certainty factors
For example, given a logical expression for combining evidence
such as:
E = (E1 AND E2 AND E3) OR (E4 AND NOT E5)
The evidence E would be computed as:
E = max [min (E1, E2, E3) ,min(E4, -E5) ]
For values:
E1 = 0.9
E4 = -0.5
E2 = 0.8
E5 = -0.4
E3 = 03
the result:
E = max[min(0.9,0.8,0.3), min(-0.5,-(-0.4)]
= max[0.3, -0.5]
= 0.3
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Calculating with certainty factors
The fundamental formula for the CF of a rule:
IF E THEN H
is given by the formula:
(8) CF(H,e) = CF(E,e) CF(H,E)
where:
CF(E,e) is the certainty factor of the evidence E making up the antecedent of
the rule based on uncertain evidence e.
CF(H,E) is the certainty factor of the hypothesis assuming that the evidence is
known with certainty, when CF(E,e) = 1.
CF(H,e) is the certainty factor of the hypothesis based on uncertain evidence e.
If all the evidence in the antecedent is known with certainty (CF(E,e) = 1)
CF(H,e) = CF(H,E)
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Calculating with certainty factors
As an example:
IF 1) The stain of the organism is gram positive,
and
2) The morphology of the organism is ooccus, and
3) The growth conformation of the organism is chains
THEN There is suggestive evidence (0.7) that the identity of the
organism is streptococcus
where the certainty factor of the hypothesis under certain evidence is :
and is also called the attenuation factor.
The attenuation factor
is based on the assumption that all the evidence E1, E2 and E3 is known with
certainty
expresses the degree of certainty of the hypothesis, given certain evidence
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Calculating with certainty factors
A complication occurs if all the evidence is not known with certainty
For example
CF (E1 , e) = 0. 5
CF (E2 , e) = 0. 6
CF (E3 , e) = 0. 3
then:
The certainty factor of the conclusion is :
CF(H,e) = CF(E,e) CF(H,E)
= 0.3 * 0.7
= 0.21
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Calculating with certainty factors
Suppose another rule also concludes the same hypothesis, but
with a different certainty factor.
The certainty factors of rules concluding the same hypothesis are
calculated from the combining function
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Calculating with certainty factors
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Calculating with certainty factors
if another rule concludes streptococcus with certainty factor CF, = 0.5,
then the combined certainty
CFCOMBINE (0.21,0.5)= 0.21 + 0.5 (1 - 0.21) = 0.605
Suppose a third rule also has the same conclusion, but with a
CF3 = - 0.4.
The CFCOMBINE formula preserves the commutativity of evidence. That
is
CFCOMBINE(X,Y) = CFCOMBINE(Y,X)
MYCIN stored the current CFCOMBINE with each hypothesis and
combined it with new evidence as it became available
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