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FT228/4 Knowledge Based Decision Support Systems . Uncertainty Management in Rule-Based Systems Certainty Factors. Ref: Artificial Intelligence A Guide to Intelligent Systems Michael Negnevitsky – Aungier St. Call No. 006.3. Uncertainty Approaches in AI. Quantitative Numerical Approaches

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Ft228 4 knowledge based decision support systems

FT228/4 Knowledge Based Decision Support Systems

Uncertainty Management in Rule-Based Systems

Certainty Factors

Ref: Artificial Intelligence A Guide to Intelligent Systems

Michael Negnevitsky – Aungier St. Call No. 006.3

Uncertainty approaches in ai
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

Arguments against probability
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

Bayesian inference
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

Certainty factors
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

Certainty factors1
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

Total strength of belief
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)]


  • 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

Certainty factors2
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}

Certainty factors3
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

Stanford certainty factor algebra
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)


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)

= -0.5 . 0.8 = -0.4

Certainty factors conjunctive rules
Certainty Factors – Conjunctive Rules

IF <evidence1>

AND <evidence2>



AND <evidencen>


<hypothesis H> {cf}

cf(H, E1  E2  …  En) =

min[cf(E1),cf(E2)…cf(En)] x cf

Certainty factors conjunctive rules1
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


    = min[0.9,0.7].0.8


Certainty factors disjunctive rules
Certainty Factors – Disjunctive Rules

IF <evidence1>

OR <evidence2>



OR <evidencen>


<hypothesis H> {cf}

cf(H, E1  E2  …  En) =

max[cf(E1),cf(E2)…cf(En)] x cf

Certainty factors disjunctive rules1
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


    = max[0.6,0.8].0.8


Consequent from multiple rules
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 + cf2 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

Consequent from multiple rules1
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)]

    = 0.8 + 0.6 x(1 –0.8)= 0.92

Certainty factors4
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

Certainty factors problems
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

Comparison of bayesian reasoning certainty factors
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

Comparison of bayesian reasoning certainty factors1
Comparison of Bayesian Reasoning & Certainty Factors

  • 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