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FT228/4 Knowledge Based Decision Support Systems PowerPoint Presentation

FT228/4 Knowledge Based Decision Support Systems

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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
- Probability Theory
- Certainty Factors
- Dempster-Shafer evidential theory
- Fuzzy logic

- Numerical Approaches
- Qualitative
- Logical Approaches
- Reasoning by cases
- Non-monotonic reasoning

- Logical Approaches
- Hybrid approaches

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

- 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 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 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

- 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)]

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

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 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

- 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)

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)

= -0.5 . 0.8 = -0.4

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

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

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

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

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 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 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

- 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

- 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 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

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