FT228/4 Knowledge Based Decision Support Systems

Uncertainty Management in Rule-Based SystemsCertainty 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

Qualitative Logical Approaches

Reasoning by cases Non-monotonic reasoning

Hybrid approaches

Arguments against probability

Requires massive amount of dataRequires enumeration of all possibilitiesHides details of character of uncertaintyPeople are bad probability estimatorsDifficult to use

Bayesian InferenceDescribes the application domain as a set of possible outcomes termed hypothesesRequires an initial probability for each hypothesis in the problem space Prior probability

Bayesian inference then updates probabilities using evidenceEach 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 xRange from –1 to +1 cf=-1 very uncertain cf=+1 very certain cf=0 neutral

Certainty factors are relative measuresDo not translate to measure of absolute belief

Total Strength of Belief

Certainty factors combin belief and disbelief into a single number based on some evidenceMB(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 > MDCF of 1 means evidence definitely supports the hypothesisCF of 0 means either there is no evidence or that the belief is cancelled out by the disbeliefNegative CF implies that the evidence favours negation of hypothesis since MB < MD

Certainty Factors

Consider a simple ruleIF 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 chainEstablishes 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 ruleIf x has wings then x is a birdLet the cf of this rule be 0.8IF (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 exampleIF sky is clear AND forecast is sunnyTHEN wear sunglasses cf{0.8}cf(sky is clear)=0.9cf(forecast is sunny)=0.7cf(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 exampleIF sky is overcast AND forecast is rainTHEN take umbrella cf{0.9}cf(sky is overcast)=0.6cf(forecast is rain)=0.8cf(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 THENC 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 < 0cf1=confidence in hypothesis established by Rule 1cf2=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)=1.0cf1(H,E1)=cf(E1) x cf = 1.0 x 0.8 = 0.8cf2(H,E2)=cf(E2) x cf = 1.0 x 0.6 = 0.6Cf(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 reasoningHeuristic manner of combining certainty factors differs from the way in which they would be combined if they were probabilitiesNot mathematically pureDoes mimic thinking process of human expert

Certainty Factors - Problems

Results may depend on order in which evidence considered in some casesReasoning often fairly insensitive to themDon’t capture credibility in some casesWhat 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