1. FT228/4 Knowledge Based Decision Support Systems
Uncertainty Management in Rule-Based Systems
Certainty Factors
More on inference chaining
Ref: Artificial Intelligence A Guide to Intelligent Systems
Michael Negnevitsky – Aungier St. Call No

2. Uncertainty Approaches in AI
Quantitative
Numerical Approaches
Probability Theory
Certainty Factors
Dempster-Shafer evidential theory
Fuzzy logic
Qualitative
Logical Approaches
Reasoning by cases
Non-monotonic reasoning
Hybrid approaches
Probability theory rejected because of too many numbers, irrelevant to commonsense reasoning
Rehabilitated using semi-qualitative representations (Bayes nets)

3. Arguments against probability
Requires massive amount of data
Requires enumeration of all possibilities
Hides details of character of uncertainty
People are bad probability estimators
Difficult to use
Probability is a scientific measure of chance
The proportion of cases in which an event occurs
Probability can be expressed as a numerical index with a range 0 to unity

4. Bayesian Inference
Describes the application domain as a set of possible outcomes termed hypotheses
Requires an initial probability for each hypothesis in the problem space
Prior probability
Bayesian inference then updates probabilities using evidence
Each piece of evidence may update the probability of a set of hypotheses
Represent revised beliefs in light of known evidence
Mathematically calculated from Bayes theorem

5. Certainty Factors Certainty factors express belief in an event
Fact or hypothesis
Based upon evidence
Experts assessment
Composite number that can be used to
Guide reasoning
Cause a current goal to be deemed unpromising and pruned from search space
Rank hypotheses after all evidence has been considered

6. Certainty Factors
Certainty Factor cf(x) is a measure of how confident we are in x
Range from –1 to +1
cf=-1 very uncertain
cf=+1 very certain
cf=0 neutral
Certainty factors are relative measures
Do not translate to measure of absolute belief
First introduced in an expert system called MYCIN for the diagnosis and therapy of blood infections

7. Total Strength of Belief
Certainty factors combin belief and disbelief into a single number based on some evidence
MB(H,E)
MD(H,E)
Strength of belief or disbelief in H depends on the kind of evidence E observed
cf= MB(H,E) – MD(H,E)
1 – min[MB(H,E), MD(H,E)]
Some facts increase belief some facts increase disbelief

8. Belief
Positive CF implies evidence supports hypothesis since MB > MD
CF of 1 means evidence definitely supports the hypothesis
CF of 0 means either there is no evidence or that the belief is cancelled out by the disbelief
Negative CF implies that the evidence favours negation of hypothesis since MB < MD

9. Certainty Factors Consider a simple rule IF A is X THEN B is Y
Expert may not be absolutely certain rule holds
Suppose it has been observed that in some cases even when the antecedent is true, A takes value X, the consequent is false and B takes a different value Z
IF A is X THEN B is Y {cf 0.7};
B is Z {cf 0.2}

10. Certainty Factors
Factor assigned by the rule is propagated through the reasoning chain
Establishes the net certainty of the consequent when the evidence for the antecedent is uncertain

11. Stanford Certainty Factor Algebra
There are rules to combine CFs of several facts
(cf(x1) AND cf(x2)) = min(cf(x1),cf(x2))
(cf(x1) OR cf(x2)) = max(cf(x1),cf(x2))
A rule may also have a certainty factor cf(rule)
cf(action) = cf(condition).cf(rule)

12. Example cf(shep is a dog)=0.7 cf(shep has wings)=-0.5
cf(Shep is a dog and has wings) = min(0.7, -0.5)
= -0.5
Suppose there is a rule
If x has wings then x is a bird
Let the cf of this rule be 0.8
IF (Shep has wings) then (Shep is a bird)
= = -0.4

13. Certainty Factors – Conjunctive Rules
IF <evidence1>
AND <evidence2> .
AND <evidencen>
THEN
<hypothesis H> {cf}
cf(H, E1  E2  …  En) =
min[cf(E1),cf(E2)…cf(En)] x cf

14. Certainty Factors – Conjunctive Rules
For example
IF sky is clear AND forecast is sunny
THEN wear sunglasses cf{0.8}
cf(sky is clear)=0.9
cf(forecast is sunny)=0.7
cf(action)=cf(condition).cf(rule)
= min[0.9,0.7].0.8
=0.56

15. Certainty Factors – Disjunctive Rules
IF <evidence1>
OR <evidence2> .
OR <evidencen>
THEN
<hypothesis H> {cf}
cf(H, E1  E2  …  En) =
max[cf(E1),cf(E2)…cf(En)] x cf

16. Certainty Factors – Disjunctive Rules
For example
IF sky is overcast AND forecast is rain
THEN take umbrella cf{0.9}
cf(sky is overcast)=0.6
cf(forecast is rain)=0.8
cf(action)=cf(condition).cf(rule)
= max[0.6,0.8].0.8
=0.72

17. Consequent from multiple rules
Suppose we have the following :
IF A is X THEN C is Z {cf 0.8}
IF B is Y THEN C is Z {cf 0.6}
What certainty should be attached to C having Z if both rules are fired ?
cf(cf1,cf2)= cf1 + cf2 x (1- cf1) if cf1> 0 and cf2 > 0
= cf1 + cf if cf1 < 0 orcf2 < 0
1- min[|cf1|,|cf2|]
= cf1+cf2 x (1+cf1) if cf1 < 0 and cf2 < 0
cf1=confidence in hypothesis established by Rule 1
cf2=confidence in hypothesis established by Rule 2
|cf1| and |cf2| are absolute magnitudes of cf1 and cf2
Common sense would say that our confidence in the hypothesis should increase if was have more evidence from different sources

18. Consequent from multiple rules
cf(E1)=cf(E2)=1.0
cf1(H,E1)=cf(E1) x cf = 1.0 x 0.8 = 0.8
cf2(H,E2)=cf(E2) x cf = 1.0 x 0.6 = 0.6
Cf(cf1,cf2)= cf1(H,E1) + cf2(H,E2) x [1-cf1(H,E1)]
= x(1 –0.8)= 0.92

19. Certainty Factors Practical alternative to Bayesian reasoning
Heuristic manner of combining certainty factors differs from the way in which they would be combined if they were probabilities
Not mathematically pure
Does mimic thinking process of human expert

20. Certainty Factors - Problems
Results may depend on order in which evidence considered in some cases
Reasoning often fairly insensitive to them
Don’t capture credibility in some cases
What do they mean exactly ?
In some cases can be interpreted probabilistically

21. Comparison of Bayesian Reasoning & Certainty Factors
Probability Theory
Oldest & best-established technique
Works well in areas such as forecasting & planning
Areas where statistical data is available and probability statements made
Most expert system application areas do not have reliable statistical information
Assumption of conditional independence cannot be made
Leads to dissatisfaction with method

22. Comparison of Bayesian Reasoning & Certainty Factors
Lack mathematical correctness of probability theory
Outperforms Bayesian reasoning in areas such as diagnostics and particularly medicine
Used in cases where probabilities are not known or too difficult or expensive to obtain
Evidential reasoning
Can manage incrementally acquired evidence
Conjunction and disjunction of hypotheses
Evidences with varying degree of belief
Provide better explanations of control flow