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 • 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 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

FT228/4 Knowledge Based Decision Support Systems