Knowledge-Based Knowledge-Based

SystemsSystems

Knowledge-Based Knowledge-Based

SystemsSystemsDr. Marco Antonio Ramos CorchadoDr. Marco Antonio Ramos Corchado

Computer Science DepartmentComputer Science Department

Overview Reasoning Overview Reasoning and Uncertaintyand Uncertainty

Overview Reasoning Overview Reasoning and Uncertaintyand Uncertainty

MotivationMotivation

ObjectivesObjectives

Sources of Uncertainty and Sources of Uncertainty and Inexactness in ReasoningInexactness in Reasoning Incorrect and Incomplete Incorrect and Incomplete

KnowledgeKnowledge AmbiguitiesAmbiguities Belief and IgnoranceBelief and Ignorance

Probability TheoryProbability Theory Bayesian NetworksBayesian Networks

Certainty FactorsCertainty Factors Belief and DisbeliefBelief and Disbelief

Dempster-Shafer TheoryDempster-Shafer Theory Evidential ReasoningEvidential Reasoning

Important Concepts and Important Concepts and TermsTerms

Chapter SummaryChapter Summary

LogisticsLogisticsLogisticsLogistics

IntroductionsIntroductions

Course MaterialsCourse Materials textbooks (see below)textbooks (see below) lecture noteslecture notes

PowerPoint Slides will be available on my Web pagePowerPoint Slides will be available on my Web page handoutshandouts Web pageWeb page

http://fi.uaemex.mx/mramoshttp://fi.uaemex.mx/mramos

Term ProjectTerm Project

Lab and Homework AssignmentsLab and Homework Assignments

ExamsExams

GradingGrading

Bridge-InBridge-In

Pre-TestPre-Test

MotivationMotivationMotivationMotivation

• reasoning for real-world problems involves missing reasoning for real-world problems involves missing knowledge, inexact knowledge, inconsistent facts or knowledge, inexact knowledge, inconsistent facts or rules, and other sources of uncertaintyrules, and other sources of uncertainty

• while traditional logic in principle is capable of while traditional logic in principle is capable of capturing and expressing these aspects, it is not capturing and expressing these aspects, it is not very intuitive or practicalvery intuitive or practical• explicit introduction of predicates or functionsexplicit introduction of predicates or functions

• many expert systems have mechanisms to deal with many expert systems have mechanisms to deal with uncertaintyuncertainty• sometimes introduced as ad-hoc measures, lacking a sometimes introduced as ad-hoc measures, lacking a

sound foundationsound foundation

ObjectivesObjectivesObjectivesObjectives

• be familiar with various sources of uncertainty and imprecision be familiar with various sources of uncertainty and imprecision in knowledge representation and reasoningin knowledge representation and reasoning

• understand the main approaches to dealing with uncertaintyunderstand the main approaches to dealing with uncertainty• probability theoryprobability theory

• Bayesian networksBayesian networks• Dempster-Shafer theoryDempster-Shafer theory

• important characteristics of the approachesimportant characteristics of the approaches• differences between methods, advantages, disadvantages, performance, differences between methods, advantages, disadvantages, performance,

typical scenariostypical scenarios

• evaluate the suitability of those approachesevaluate the suitability of those approaches• application of methods to scenarios or tasksapplication of methods to scenarios or tasks

• apply selected approaches to simple problemsapply selected approaches to simple problems

IntroductionIntroduction

• reasoning under uncertainty and with inexact knowledgereasoning under uncertainty and with inexact knowledge• frequently necessary for real-world problemsfrequently necessary for real-world problems

• heuristicsheuristics• ways to mimic heuristic knowledge processingways to mimic heuristic knowledge processing• methods used by expertsmethods used by experts

• empirical associationsempirical associations• experiential reasoningexperiential reasoning• based on limited observationsbased on limited observations

• probabilitiesprobabilities• objective (frequency counting)objective (frequency counting)• subjective (human experience )subjective (human experience )

• reproducibilityreproducibility• will observations deliver the same results when repeatedwill observations deliver the same results when repeated

Dealing with Dealing with UncertaintyUncertainty

• expressivenessexpressiveness• can concepts used by humans be represented adequately?can concepts used by humans be represented adequately?• can the confidence of experts in their decisions be expressed?can the confidence of experts in their decisions be expressed?

• comprehensibilitycomprehensibility• representation of uncertaintyrepresentation of uncertainty• utilization in reasoning methodsutilization in reasoning methods

• correctnesscorrectness• probabilitiesprobabilities

• adherence to the formal aspects of probability theory adherence to the formal aspects of probability theory

• relevance rankingrelevance ranking• probabilities don’t add up to 1, but the “most likely” result is sufficientprobabilities don’t add up to 1, but the “most likely” result is sufficient

• long inference chainslong inference chains• tend to result in extreme (0,1) or not very useful (0.5) resultstend to result in extreme (0,1) or not very useful (0.5) results

• computational complexitycomputational complexity• feasibility of calculations for practical purposesfeasibility of calculations for practical purposes

Sources of Sources of UncertaintyUncertainty

• datadata• data missing, unreliable, ambiguous, data missing, unreliable, ambiguous, • representation imprecise, inconsistent, subjective, derived from representation imprecise, inconsistent, subjective, derived from

defaults, …defaults, …

• expert knowledgeexpert knowledge• inconsistency between different expertsinconsistency between different experts• plausibilityplausibility

• ““best guess” of expertsbest guess” of experts• qualityquality

• causal knowledgecausal knowledge• deep understandingdeep understanding

• statistical associationsstatistical associations• observationsobservations

• scopescope• only current domain, or more generalonly current domain, or more general

Sources of Sources of Uncertainty (cont.)Uncertainty (cont.)

• knowledge representationknowledge representation• restricted model of the real systemrestricted model of the real system• limited expressiveness of the representation mechanismlimited expressiveness of the representation mechanism

• inference processinference process• deductivedeductive

• the derived result is formally correct, but inappropriatethe derived result is formally correct, but inappropriate• derivation of the result may take very longderivation of the result may take very long

• inductiveinductive• new conclusions are not well-foundednew conclusions are not well-founded

• not enough samplesnot enough samples• samples are not representativesamples are not representative

• unsound reasoning methodsunsound reasoning methods• induction, non-monotonic, default reasoninginduction, non-monotonic, default reasoning

Uncertainty in Uncertainty in Individual RulesIndividual Rules

• errorserrors• domain errorsdomain errors

• representation errorsrepresentation errors

• inappropriate application of the ruleinappropriate application of the rule

• likelihood of evidencelikelihood of evidence• for each premisefor each premise

• for the conclusionfor the conclusion

• combination of evidence from multiple premisescombination of evidence from multiple premises

Uncertainty and Uncertainty and Multiple RulesMultiple Rules

• conflict resolutionconflict resolution• if multiple rules are applicable, which one is selectedif multiple rules are applicable, which one is selected

• explicit priorities, provided by domain expertsexplicit priorities, provided by domain experts• implicit priorities derived from rule propertiesimplicit priorities derived from rule properties

• specificity of patterns, ordering of patterns creation time of rules, specificity of patterns, ordering of patterns creation time of rules, most recent usage, …most recent usage, …

• compatibilitycompatibility• contradictions between rulescontradictions between rules• subsumptionsubsumption

• one rule is a more general version of another oneone rule is a more general version of another one• redundancyredundancy• missing rulesmissing rules• data fusiondata fusion

• integration of data from multiple sourcesintegration of data from multiple sources

Basics of Basics of Probability TheoryProbability Theory

• mathematical approach for processing uncertain informationmathematical approach for processing uncertain information

• sample space setsample space setX = {xX = {x11, x, x22, …, x, …, xnn}}• collection of all possible eventscollection of all possible events• can be discrete or continuouscan be discrete or continuous

• probability number P(xprobability number P(xii) reflects the likelihood of an event x) reflects the likelihood of an event x ii to to occuroccur• non-negative value in [0,1]non-negative value in [0,1]• total probability of the sample space (sum of probabilities) is 1total probability of the sample space (sum of probabilities) is 1• for mutually exclusive events, the probability for at least one of them is the for mutually exclusive events, the probability for at least one of them is the

sum of their individual probabilitiessum of their individual probabilities• experimental probabilityexperimental probability

• based on the frequency of eventsbased on the frequency of events• subjective probabilitysubjective probability

• based on expert assessmentbased on expert assessment

Compound Compound ProbabilitiesProbabilities

• describes describes independentindependent events events• do not affect each other in any waydo not affect each other in any way

• jointjoint probability of two independent events A and B probability of two independent events A and B P(A P(A B) B) = n(A = n(A B) / n(s) = P(A) * P (B) B) / n(s) = P(A) * P (B)

where n(S) is the number of elements in Swhere n(S) is the number of elements in S

• unionunion probability of two independent events A and B probability of two independent events A and BP(A P(A B) B) = P(A) + P(B) - P(A = P(A) + P(B) - P(A B) B)

= P(A) + P(B) - P(A) * P (B)= P(A) + P(B) - P(A) * P (B)

Conditional Conditional ProbabilitiesProbabilities

• describes describes dependentdependent events events• affect each other in some wayaffect each other in some way

• conditional probabilityconditional probability of event A given that event B has already occurredof event A given that event B has already occurredP(A|B)P(A|B) = P(A = P(A B) / P(B) B) / P(B)

Advantages and Problems: Advantages and Problems: ProbabilitiesProbabilities

• advantagesadvantages• formal foundationformal foundation• reflection of reality (a posteriori)reflection of reality (a posteriori)

• problemsproblems• may be inappropriatemay be inappropriate

• the future is not always similar to the pastthe future is not always similar to the past• inexact or incorrectinexact or incorrect

• especially for subjective probabilitiesespecially for subjective probabilities• ignoranceignorance

• probabilities must be assigned even if no information is availableprobabilities must be assigned even if no information is available• assigns an equal amount of probability to all such itemsassigns an equal amount of probability to all such items

• non-local reasoningnon-local reasoning• requires the consideration of all available evidence, not only from the rules requires the consideration of all available evidence, not only from the rules

currently under considerationcurrently under consideration• no compositionalityno compositionality

• complex statements with conditional dependencies can not be decomposed into complex statements with conditional dependencies can not be decomposed into independent partsindependent parts

Bayesian ApproachesBayesian Approaches

• derive the probability of a cause given a symptomderive the probability of a cause given a symptom

• has gained importance recently due to advances in has gained importance recently due to advances in efficiencyefficiency• more computational power availablemore computational power available• better methodsbetter methods

• especially useful in diagnostic systemsespecially useful in diagnostic systems• medicine, computer help systemsmedicine, computer help systems

• inverseinverse or or a posterioria posteriori probability probability• inverse to conditional probability of an earlier event given inverse to conditional probability of an earlier event given

that a later one occurredthat a later one occurred

Bayes’ Rule for Bayes’ Rule for Single EventSingle Event

• single hypothesis H, single event Esingle hypothesis H, single event EP(H|E) = (P(E|H) * P(H)) / P(E)P(H|E) = (P(E|H) * P(H)) / P(E)oror

• P(H|E) = (P(E|H) * P(H) / P(H|E) = (P(E|H) * P(H) / (P(E|H) * P(H) + P(E| (P(E|H) * P(H) + P(E|H) * P(H) * P(H) )H) )

Bayes’ Rule for Bayes’ Rule for Multiple EventsMultiple Events

• multiple hypotheses Hmultiple hypotheses Hii, multiple events E, multiple events E11, …, E, …, Enn

P(HP(Hii|E|E11, E, E22, …, E, …, Enn) )

= (P(E= (P(E11, E, E22, …, E, …, Enn|H|Hii) * P(H) * P(Hii)) / P(E)) / P(E11, E, E22, …, , …,

EEnn))

ororP(HP(Hii|E|E11, E, E22, …, E, …, Enn) )

= (P(E= (P(E11|H|Hii) * P(E) * P(E22|H|Hii) * …* P(E) * …* P(Enn|H|Hii) * P(H) * P(Hii)) / )) /

kk P(E P(E11|H|Hkk) * P(E) * P(E22|H|Hkk) * … * P(E) * … * P(Enn|H|Hkk)* )*

P(HP(Hkk))

with independent pieces of evidence E with independent pieces of evidence Eii

Advantages and Problems of Advantages and Problems of Bayesian ReasoningBayesian Reasoning

• advantagesadvantages• sound theoretical foundationsound theoretical foundation• well-defined semantics for decision makingwell-defined semantics for decision making

• problemsproblems• requires large amounts of probability datarequires large amounts of probability data

• sufficient sample sizessufficient sample sizes• subjective evidence may not be reliablesubjective evidence may not be reliable• independence of evidences assumption often not validindependence of evidences assumption often not valid• relationship between hypothesis and evidence is reduced to a relationship between hypothesis and evidence is reduced to a

numbernumber• explanations for the user difficultexplanations for the user difficult• high computational overheadhigh computational overhead

Certainty FactorsCertainty Factors

• denotes the belief in a hypothesis H given that some denotes the belief in a hypothesis H given that some pieces of evidence E are observedpieces of evidence E are observed

• no statements about the belief means that no evidence is no statements about the belief means that no evidence is presentpresent• in contrast to probabilities, Bayes’ methodin contrast to probabilities, Bayes’ method

• works reasonably well with partial evidenceworks reasonably well with partial evidence• separation of belief, disbelief, ignoranceseparation of belief, disbelief, ignorance

• share some foundations with Dempster-Shafer theory, share some foundations with Dempster-Shafer theory, but are more practicalbut are more practical• introduced in an ad-hoc way in MYCINintroduced in an ad-hoc way in MYCIN• later mapped to DS theorylater mapped to DS theory

Belief and DisbeliefBelief and Disbelief

• measure of beliefmeasure of belief• degree to which hypothesis H is supported by degree to which hypothesis H is supported by

evidence Eevidence E

• MB(H,E) = 1 if P(H) =1MB(H,E) = 1 if P(H) =1 (P(H|E) - P(H)) / (1- P(H)) otherwise (P(H|E) - P(H)) / (1- P(H)) otherwise

• measure of disbeliefmeasure of disbelief• degree to which doubt in hypothesis H is supported by degree to which doubt in hypothesis H is supported by

evidence Eevidence E

• MB(H,E) = 1 if P(H) =0MB(H,E) = 1 if P(H) =0 (P(H) - P(H|E)) / P(H)) otherwise (P(H) - P(H|E)) / P(H)) otherwise

Certainty FactorCertainty Factor

• certainty factor CF certainty factor CF • ranges between -1 (denial of the hypothesis H) and +1 ranges between -1 (denial of the hypothesis H) and +1

(confirmation of H)(confirmation of H)• allows the ranking of hypothesesallows the ranking of hypotheses

• difference between belief and disbeliefdifference between belief and disbeliefCF (H,E) = MB(H,E) - MD (H,E)CF (H,E) = MB(H,E) - MD (H,E)

• combining antecedent evidencecombining antecedent evidence• use of premises with less than absolute confidenceuse of premises with less than absolute confidence• EE11 E E22 = min(CF(H, E = min(CF(H, E11), CF(H, E), CF(H, E22))))• EE11 E E22 = max(CF(H, E = max(CF(H, E11), CF(H, E), CF(H, E22))))• E = E = CF(H, E) CF(H, E)

Combining Certainty Combining Certainty FactorsFactors

• certainty factors that support the same conclusioncertainty factors that support the same conclusion

• several rules can lead to the same conclusionseveral rules can lead to the same conclusion

• applied incrementally as new evidence becomes availableapplied incrementally as new evidence becomes available

CFCFrevrev(CF(CFoldold, CF, CFnewnew) = ) =

CFCFoldold + CF + CFnewnew(1 - CF(1 - CFoldold) ) if both > 0if both > 0

CFCFoldold + CF + CFnewnew(1 + CF(1 + CFoldold) ) if both < 0if both < 0

CFCFoldold + CF + CFnewnew / (1 - min(|CF / (1 - min(|CFoldold|, |CF|, |CFnewnew|)) |)) if one < 0if one < 0

Characteristics of Characteristics of Certainty FactorsCertainty Factors

RangesRangesmeasure of beliefmeasure of belief 0 ≤ 0 ≤ MBMB ≤ 1 ≤ 1measure of disbeliefmeasure of disbelief 0 ≤ 0 ≤ MDMD ≤ 1 ≤ 1certainty factor certainty factor -1 ≤ -1 ≤ CFCF ≤ +1 ≤ +1

Aspect Probability MB MD CF

Certainly true P(H|E) = 1 1 0 1

Certainly false P(H|E) = 1 0 1 -1

No evidence P(H|E) = P(H) 0 0 0

Advantages and Problems of Advantages and Problems of Certainty FactorsCertainty Factors

• AdvantagesAdvantages• simple implementationsimple implementation• reasonable modeling of human experts’ beliefreasonable modeling of human experts’ belief

• expression of belief and disbeliefexpression of belief and disbelief• successful applications for certain problem classessuccessful applications for certain problem classes• evidence relatively easy to gatherevidence relatively easy to gather

• no statistical base requiredno statistical base required

• ProblemsProblems• partially ad hoc approachpartially ad hoc approach

• theoretical foundation through Dempster-Shafer theory was developed latertheoretical foundation through Dempster-Shafer theory was developed later• combination of non-independent evidence unsatisfactorycombination of non-independent evidence unsatisfactory• new knowledge may require changes in the certainty factors of existing new knowledge may require changes in the certainty factors of existing

knowledgeknowledge• certainty factors can become the opposite of conditional probabilities for certain certainty factors can become the opposite of conditional probabilities for certain

casescases• not suitable for long inference chainsnot suitable for long inference chains

Dempster-Shafer Dempster-Shafer TheoryTheory

• mathematical theory of evidencemathematical theory of evidence• uncertainty is modeled through a range of probabilitiesuncertainty is modeled through a range of probabilities• instead of a single number indicating a probabilityinstead of a single number indicating a probability

• sound theoretical foundationsound theoretical foundation

• allows distinction between belief, disbelief, ignorance allows distinction between belief, disbelief, ignorance (non-belief)(non-belief)

• certainty factors are a special case of DS theorycertainty factors are a special case of DS theory

DS Theory NotationDS Theory Notation

• environment environment = {O = {O11, O, O22, ..., O, ..., Onn}}• set of objects Oset of objects Oii that are of interest that are of interest• = {O= {O11, O, O22, ..., O, ..., Onn}}

• frame of discernment FDframe of discernment FD• an environment whose elements may be possible answersan environment whose elements may be possible answers• only one answer is the correct oneonly one answer is the correct one

• mass probability function mmass probability function m• assigns a value from [0,1] to every item in the frame of discernmentassigns a value from [0,1] to every item in the frame of discernment• describes the degree of belief in analogy to the mass of a physical describes the degree of belief in analogy to the mass of a physical

objectobject

• mass probabilitymass probability m(A) m(A)• portion of the total mass probability that is assigned to a specific portion of the total mass probability that is assigned to a specific

element A of FDelement A of FD

Belief and CertaintyBelief and Certainty

• belief Bel(A) in a subset A belief Bel(A) in a subset A • sum of the mass probabilities of all the proper subsets sum of the mass probabilities of all the proper subsets

of Aof A

• likelihood that one of its members is the conclusionlikelihood that one of its members is the conclusion

• plausibility Pl(A)plausibility Pl(A)• maximum belief of Amaximum belief of A

• certainty Cer(A)certainty Cer(A)• interval [Bel(A), Pl(A)]interval [Bel(A), Pl(A)]

• expresses the range of beliefexpresses the range of belief

Combination of Mass Combination of Mass ProbabilitiesProbabilities

• combining two masses in such a way that the new mass combining two masses in such a way that the new mass represents a consensus of the contributing pieces of evidencerepresents a consensus of the contributing pieces of evidence• set intersection puts the emphasis on common elements of set intersection puts the emphasis on common elements of

evidence, rather than conflicting evidence evidence, rather than conflicting evidence

• mm11 m m22 (C) (C) = = X X Y Y m m11(X) * m(X) * m22(Y)(Y)

=C m=C m11(X) * m(X) * m22(Y) / (1- (Y) / (1- X X Y) Y)=C m=C m11(X) * m(X) * m22(Y) (Y)

where where X, Y are hypothesis subsets X, Y are hypothesis subsets and and

C is their intersection C = X C is their intersection C = X Y Y

is the orthogonal or direct is the orthogonal or direct sumsum

Differences Differences Probabilities - DF Probabilities - DF

TheoryTheory

Aspect Probabilities Dempster-Shafer

Aggregate Sum i Pi = 1 m() ≤ 1

Subset X Y P(X) ≤ P(Y) m(X) > m(Y) allowed

relationship X, X(ignorance)

P(X) + P (X) = 1 m(X) + m(X) ≤ 1

Evidential Reasoning Evidential Reasoning

• extension of DS theory that deals with uncertain, extension of DS theory that deals with uncertain, imprecise, and possibly inaccurate knowledgeimprecise, and possibly inaccurate knowledge

• also uses evidential intervals to express the also uses evidential intervals to express the confidence in a statementconfidence in a statement

Evidential IntervalsEvidential Intervals

BelBel: belief; lower bound of the evidential interval: belief; lower bound of the evidential interval

PlsPls: plausibility; upper bound: plausibility; upper bound

Meaning Evidential Interval

Completely true [1,1]

Completely false [0,0]

Completely ignorant [0,1]

Tends to support [Bel,1] where 0 < Bel < 1

Tends to refute [0,Pls] where 0 < Pls < 1

Tends to both support and refute [Bel,Pls] where 0 < Bel ≤ Pls< 1

Advantages and Problems of Advantages and Problems of Dempster-ShaferDempster-Shafer

• advantagesadvantages• clear, rigorous foundationclear, rigorous foundation

• ability to express confidence through intervalsability to express confidence through intervals

• certainty about certaintycertainty about certainty

• proper treatment of ignoranceproper treatment of ignorance

• problemsproblems• non-intuitive determination of mass probabilitynon-intuitive determination of mass probability

• very high computational overheadvery high computational overhead

• may produce counterintuitive results due to normalizationmay produce counterintuitive results due to normalization

• usability somewhat unclearusability somewhat unclear

Post-TestPost-Test

Important Concepts Important Concepts and Termsand Terms

Important Concepts Important Concepts and Termsand Terms

• Bayesian networksBayesian networks

• beliefbelief

• certainty factorcertainty factor

• compound probabilitycompound probability

• conditional probabilityconditional probability

• Dempster-Shafer theoryDempster-Shafer theory

• disbeliefdisbelief

• evidential reasoningevidential reasoning

• inferenceinference

• inference mechanisminference mechanism

• ignoranceignorance

• knowledgeknowledge

• knowledge knowledge representationrepresentation

• mass functionmass function

• probabilityprobability

• reasoningreasoning

• rulerule

• samplesample

• setset

• uncertaintyuncertainty

Summary Reasoning Summary Reasoning and Uncertaintyand Uncertainty

Summary Reasoning Summary Reasoning and Uncertaintyand Uncertainty

• many practical tasks require reasoning under many practical tasks require reasoning under uncertaintyuncertainty• missing, inexact, inconsistent knowledgemissing, inexact, inconsistent knowledge

• variations of probability theory are often combined variations of probability theory are often combined with rule-based approacheswith rule-based approaches• works reasonably well for many practical problemsworks reasonably well for many practical problems

• Bayesian networks have gained some prominenceBayesian networks have gained some prominence• improved methods, sufficient computational powerimproved methods, sufficient computational power