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. 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 Probability theory rejected because of too many numbers, irrelevant to commonsense reasoning Rehabilitated using semi-qualitative representations (Bayes nets)

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

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 First introduced in an expert system called MYCIN for the diagnosis and therapy of blood infections

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

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 Common sense would say that our confidence in the hypothesis should increase if was have more evidence from different sources

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 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