Uncertainty in Uncertainty in

Knowledge SystemsKnowledge Systems

IntroductionIntroduction

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fuzzyfuzzy

uncertainuncertain

possiblepossible

probableprobable

likelylikely

assumingassuming

vaguevague

Levels of certainty on the uncertainLevels of certainty on the uncertain

impreciseimprecise

approximatingapproximating

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Overview:Overview:ContentsContents ContentsContents

Inference under uncertainty:Inference under uncertainty: Abductive reasoning Abductive reasoning Probabilistic reasoningProbabilistic reasoning GenInferGenInfer Opinion Nets Opinion Nets

Sources of Uncertainty.Sources of Uncertainty.

What is a probability?What is a probability?

Diagnosis and weak implications.Diagnosis and weak implications.

Quantification types for uncertainty.Quantification types for uncertainty.

Prior versus Conditional probability.Prior versus Conditional probability.

Introducing Fuzzy sets.Introducing Fuzzy sets.

Joint probabilistic distributions.Joint probabilistic distributions. Motivating Bayes ruleMotivating Bayes rule

Axioms of probability.Axioms of probability.

Probabilistic rules.Probabilistic rules.

Utility and decision theory.Utility and decision theory.

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Sources of uncertainty:Sources of uncertainty:

1. Information obtained may take the form of1. Information obtained may take the form of weak implicationsweak implications

Ex.:Ex.: in diagnosis:in diagnosis:

disease(p, Cavity) disease(p, Cavity) (0.8)(0.8) symptom(p, Toothache) symptom(p, Toothache)

quantification of frequency quantification of frequency with which the rule applieswith which the rule applies

2. Imprecise language:2. Imprecise language:

““often”, “sometimes”, “frequently”, “hardly ever”, …often”, “sometimes”, “frequently”, “hardly ever”, …

- need to quantify these in terms of frequency,- need to quantify these in terms of frequency,

- need to deal with proposed frequency in rules.- need to deal with proposed frequency in rules.

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Study of relation between imprecise Study of relation between imprecise language and frequency:language and frequency:

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Sources of uncertainty (2):Sources of uncertainty (2):

We observe that We observe that grass_is_wetgrass_is_wet, but have no , but have no information on information on sprinklersprinkler nor nor rainrain.. How to reason?How to reason?

3. Unknown information.3. Unknown information.

grass_is_wet grass_is_wet sprinkler_was_on sprinkler_was_ongrass_is_wet grass_is_wet rain_last_night rain_last_night

quantification of possible conclusionsquantification of possible conclusions

4. Conflicting information.4. Conflicting information.

Ex.:Ex.: Several experts have provided conflicting Several experts have provided conflicting information:information:

quantification of measure of beliefquantification of measure of belief

Abductive reasoningAbductive reasoning Note: can be Note: can be “ranges“ranges”” of unknown, depending of unknown, depending

on on additional evidence.additional evidence.

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Sources of uncertainty (3):Sources of uncertainty (3):5. Vague Concepts:5. Vague Concepts:

Herman is tall. Herman is tall.

What is Herman’s height?What is Herman’s height?

- at least - at least 1.80 m1.80 m??- could Herman be - could Herman be 1.78 m1.78 m and still tall? and still tall?- if Herman is in the population of Basketball players- if Herman is in the population of Basketball players

is Herman still tall?is Herman still tall?- if Herman would be a kid of 9 years and - if Herman would be a kid of 9 years and 1.45 m1.45 m

is Herman also tall?is Herman also tall?

We may want to quantify the We may want to quantify the degree in which Herman degree in which Herman belongs to the set of belongs to the set of ‘tall_people’‘tall_people’

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Sources of uncertainty (4):Sources of uncertainty (4):5. Precise specifications may be too complex:5. Precise specifications may be too complex:

Plan_90: leave 90 minutes before departure Plan_90: leave 90 minutes before departure

ProblemProblem: will Plan_90 succeed?: will Plan_90 succeed?again depends on unknown again depends on unknown information (traffic jam?, information (traffic jam?, earthquake?, accident?)earthquake?, accident?)

BUTBUT: enumeration of all conditions may be impossible or impractical:: enumeration of all conditions may be impossible or impractical:

Succeed(Plan_90) Succeed(Plan_90) not(car_break_down) not(car_break_down) andand

not(out_of_gas) andnot(out_of_gas) and not(accident) andnot(accident) and not(traffic_jam) andnot(traffic_jam) and ... ...

Quantification of estimated degree of success Quantification of estimated degree of success instead of specification of all conditionsinstead of specification of all conditions

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Sources of uncertainty (5):Sources of uncertainty (5):6. Propagation of uncertain information:6. Propagation of uncertain information:

Tomorrow(cold) Tomorrow(cold) (0.7)(0.7)Tomorrow(rain)Tomorrow(rain) (0.6)(0.6)

Can we easily determine the uncertainty for:Can we easily determine the uncertainty for:

Tomorrow(cold) Tomorrow(cold) Tomorrow(rain) Tomorrow(rain) (?)(?)

Or for:Or for:

Tomorrow(cold) Tomorrow(cold) Tomorrow(rain) Tomorrow(rain) (?)(?)

Not without sufficient information on the Not without sufficient information on the interdependenciesinterdependencies of the events! of the events!

In In absence of interdependenciesabsence of interdependencies: propagation : propagation of uncertain knowledge of uncertain knowledge increases uncertaintyincreases uncertainty of the conclusionsof the conclusions

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Utility theory:Utility theory:Plan_90: Plan_90: leave 90 minutes before departure leave 90 minutes before departure Plan_120: Plan_120: leave 120 minutes …leave 120 minutes …Plan_1440: Plan_1440: leave 24 hours ...leave 24 hours ...

Assume that Assume that Plan_90Plan_90 is the right thing to do: is the right thing to do: whatwhat would this mean?would this mean?

Plan_120 is Plan_120 is more likelymore likely to succeed. to succeed.Plan_1440 is Plan_1440 is practically surepractically sure to succeed. to succeed.BUTBUT:: Plan_90 attempts to Plan_90 attempts to optimize all our goalsoptimize all our goals::

- - arrive on time for the flightarrive on time for the flight- avoid long waiting at the airport- avoid long waiting at the airport- avoid getting speeding tickets for the - avoid getting speeding tickets for the

drivedrive- ...- ...

Utility theoryUtility theory is used to is used to representrepresent and and reason reason about preferences.about preferences.

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Decision theory:Decision theory:

If we have expressed preferences using If we have expressed preferences using Utility Utility TheoryTheory and we have expressed probabilities of and we have expressed probabilities of events and effects by events and effects by Probability TheoryProbability Theory::

Decision theory Decision theory == Probability theoryProbability theory ++ Utility theoryUtility theory

A system is A system is rationalrationalif it chooses the action that yields the if it chooses the action that yields the

highest highest expected utilityexpected utility, averaged over all , averaged over all probabilities of outcomesprobabilities of outcomes of the of the

action.action.

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More on diagnosis andMore on diagnosis andweak implications:weak implications:

Is it possible to capture diagnosis in hard rules ?Is it possible to capture diagnosis in hard rules ?

disease(p, Cavity) disease(p, Cavity) symptom(p, Toothache) symptom(p, Toothache)

is simply wrongis simply wrong there may be other diseases !there may be other diseases !

disease(p, Cavity) disease(p, Cavity) disease(p, GumDisease) disease(p, GumDisease) disease(p, ImpactedWisdom) disease(p, ImpactedWisdom) … …

symptom(p, Toothache) symptom(p, Toothache)

Again enumeration problem: do we know them all ?Again enumeration problem: do we know them all ?

symptom(p, Toothache) symptom(p, Toothache) disease(p, Cavity) disease(p, Cavity)

wrong againwrong again not each cavity causes pain !not each cavity causes pain !

There isThere is no correct logical rule. no correct logical rule.

The best we can do is provideThe best we can do is provide “ “a quantification of beliefa quantification of belief””

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Degree of membership:Degree of membership:

What kind of quantifications?What kind of quantifications?

Basic distinctionBasic distinction Degree of beliefDegree of belief

Degree of membershipDegree of membership

Degree of belief:Degree of belief:

Probability (respecting the axioms of Probability (respecting the axioms of probability theory)probability theory)

Given Toothache, there is 80% chance of CavityGiven Toothache, there is 80% chance of Cavity

Certainty factors (don’t respect the Certainty factors (don’t respect the axioms)axioms)Toothache gives Cavity with factor 0.7 (in [-1,1])Toothache gives Cavity with factor 0.7 (in [-1,1])

Fuzzy Logic (measures in vague Fuzzy Logic (measures in vague concepts)concepts)

Herman is tall (with 95% measure of Herman is tall (with 95% measure of belonging to the set of ‘tall people’)belonging to the set of ‘tall people’)

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What is a probability?What is a probability?

An often used intuition: An often used intuition: countingcounting

P(A)P(A) = number in the range [0,1] expressing the = number in the range [0,1] expressing the degree of belief in A.degree of belief in A.

P( randomly chosen person is Chinese) P( randomly chosen person is Chinese) = ?= ?all_peopleall_people

ChineseChinese

#(Chinese)#(Chinese) #(all_people)#(all_people)==

Interesting intuition to verify basic axioms and rules of probabilityInteresting intuition to verify basic axioms and rules of probability BUTBUT: counting is not always possible, nor desirable. : counting is not always possible, nor desirable.

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But even more often:But even more often:

What is a probability? (2)What is a probability? (2) Statistics may help:Statistics may help:

Crucial is:Crucial is: Belief and probability Belief and probability changes with new gathered changes with new gathered

information !information !

Prior probabilityPrior probability Conditional probabilityConditional probability

Count a randomlyCount a randomly selected subsetselected subset of the populationof the population determinedetermine the ratiothe ratio (e.g.: of Chinese) from this subset(e.g.: of Chinese) from this subset

taken as the probabilitytaken as the probability

A general measure of A general measure of beliefbelief on the basis ofon the basis of prior experienceprior experience intuitionintuition ... ...

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Prior versus Conditional:Prior versus Conditional:

The Chinese student storyThe Chinese student story The Bombed plane storyThe Bombed plane story

Prior Probability:Prior Probability: A A = = a randomly chosen student in some classroom is a randomly chosen student in some classroom is ChineseChinese P(P(AA)) = = 1/61/6

Conditional Probability:Conditional Probability: add information:add information: B B = = the chosen classroom is in the chosen classroom is in LeuvenLeuven P(P(A | BA | B)) = = 1/5001/500

add information:add information: C C = = the classroom is from MAIthe classroom is from MAI

P(P(A | B A | B C C)) = = 1/61/6

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Prior versus Conditional (2):Prior versus Conditional (2):The Bombed plane storyThe Bombed plane story

Prior Probability:Prior Probability:

A A = = there is a bomb on boardthere is a bomb on board P(P(AA)) = = 1/101/1066

Conditional Probability:Conditional Probability: add information:add information: A’ A’ = = there is (independently) there is (independently) anotheranother bomb on board bomb on board P(P(A | A’A | A’)) = = 1/ 101/ 101212

change A’:change A’: B B = = I bring a second bomb myselfI bring a second bomb myself

P(P(A | BA | B)) = = 1/ 101/ 1066

Prior probability (Prior probability (AA):): probability of probability of AA in absence of any other informationin absence of any other information

Conditional Probability (Conditional Probability (AA||BB):): probability of probability of AA given that we already knowgiven that we already know BB

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Toothache example:Toothache example:P(Cavity) = 0.1P(Cavity) = 0.1 10% of all individuals have a Cavity10% of all individuals have a Cavity

P(Toothache) = 0.05P(Toothache) = 0.05 5% have a Toothache5% have a Toothache

P(Cavity|Toothache) = 0.8P(Cavity|Toothache) = 0.8given that we know the individual has given that we know the individual has Toothache, there is 80% chance of him having Toothache, there is 80% chance of him having CavityCavityP(Toothache|Cavity) = 0.4P(Toothache|Cavity) = 0.4conditional probability is NOT symmetricconditional probability is NOT symmetric

P(Cavity|Toothache P(Cavity|Toothache not Gumdisease) = 0.9 not Gumdisease) = 0.9additionally given that another diagnosis is additionally given that another diagnosis is already excluded, conditional probability already excluded, conditional probability increasesincreasesP(Cavity|Toothache P(Cavity|Toothache FalseTeeth) = 0 FalseTeeth) = 0adding information does not necessarily increase adding information does not necessarily increase the probabilitythe probability

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A sensible semantics for A sensible semantics for probabilistic rules:probabilistic rules:

A probabilistic ruleA probabilistic rule

A A (factor)(factor) B1 B1 B2 B2 … … Bn Bn

should best have the semantics:should best have the semantics:

P(A| B1 P(A| B1 B2 B2 … … Bn) = Bn) = factorfactor

So:So: we simply have alternative syntax:we simply have alternative syntax:

disease(p, Cavity) disease(p, Cavity) (0.8)(0.8) symptom(p, Toothache) symptom(p, Toothache)

P(Cavity|Toothache) = P(Cavity|Toothache) = 0.80.8

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But this is NOT standard at all !But this is NOT standard at all !In probabilistic Logic Programming In probabilistic Logic Programming (Subramanian et al.)(Subramanian et al.)

A A [n1,n2][n1,n2] B B [m1,m2][m1,m2]

Means:Means: IfIf the probability that in some possible the probability that in some possible world world BB is true is between is true is between m1m1 and and m2m2

ThenThen the probability that in some the probability that in some world world AA is is true is between true is between n1n1 and and nn22

Example:Example: flip a coin, with eventsflip a coin, with events AA == headhead,, BB == tailtail

head head [0.5,0.5] [0.5,0.5] tail tail [0.5,0.5][0.5,0.5]

IfIf the probability of a world in which you get the probability of a world in which you get tailtail is is 0.50.5 ThenThen the probability of a world in which you get the probability of a world in which you get headhead is also is also 0.50.5

Notice: no world in the intersection !Notice: no world in the intersection !

In conditional probabilities semantics:In conditional probabilities semantics: head head 0.0 0.0 tail tail

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The axioms of ProbabilityThe axioms of Probability

1.1. 0 0 P(A) P(A) 1 1

2.2. P(True) = 1 , P(False) = 0 P(True) = 1 , P(False) = 0

3.3. P(A P(A B) = P(A) + P(B) - P(A B) = P(A) + P(B) - P(A B) B)

AA BBA A B B

Derived: Derived: P(P(A) = 1 - P(A)A) = 1 - P(A)

Are the major difference with Are the major difference with “certainty factor”“certainty factor” systems: systems: do NOT respect these axioms do NOT respect these axioms (Mycin: factors in range -1, 1)(Mycin: factors in range -1, 1)

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Joint probability distributionJoint probability distribution Given Given 2 properties/events2 properties/events: list the entire distribution of : list the entire distribution of all all

probability assignmentsprobability assignments to all possible combinations of to all possible combinations of truth-values for the properties/eventstruth-values for the properties/events

ToothacheToothache ToothacheToothache

CavityCavity

CavityCavity

0.040.04

0.010.01

0.060.06

0.890.89

All prior and conditional probabilities can be derived ! All prior and conditional probabilities can be derived !

P(Toothache|Cavity) = P(Toothache|Cavity) = 0.04 / 0.04 + 0.06 = 0.04 / 0.04 + 0.06 = 0.40.4

BUTBUT: gathering this distribution is often not possible or at least very tedious.: gathering this distribution is often not possible or at least very tedious.

Bayes Rule !Bayes Rule !

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Inference under Uncertainty.Inference under Uncertainty.The The logic versionlogic version of the burglary-alarm example: of the burglary-alarm example:

burglaryburglary earthquakeearthquake

alarmalarm

MaryCallsMaryCallsJohnCallsJohnCalls

phoneRingsphoneRings

unlessLoudmusicunlessLoudmusic

In Logic:In Logic:

JohnCalls JohnCalls Alarm AlarmJohnCalls JohnCalls PhoneRings PhoneRingsMaryCalls MaryCalls Alarm Alarm LoudmusicLoudmusic

Alarm Alarm Burglary BurglaryAlarm Alarm EarthQuake EarthQuake

What can we deduce from an observation that What can we deduce from an observation that John callsJohn calls,, Mary doesn’t Mary doesn’t andand Mary’s CD-player was Mary’s CD-player was broken broken ??

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Abductive ReasoningAbductive Reasoning Deductive reasoning:Deductive reasoning: (using Modus Ponens)(using Modus Ponens)

AAB B A A

BB

Abductive reasoning:Abductive reasoning: (assume (assume AA is unknown) is unknown)BBB B A A

AA

AbduceAbduce that that AA holds as an explanation for the observation holds as an explanation for the observation BB More generally:More generally: given given a set of observationsa set of observations and and a set of logical rulesa set of logical rules: find : find a set of a set of

hypotheseshypotheses (from the unknown properties) that allow to deduce the observations (from the unknown properties) that allow to deduce the observations

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Abduction in burglar-alarm:Abduction in burglar-alarm:

burglaryburglary earthquakeearthquake

alarmalarm

MaryCallsMaryCallsJohnCallsJohnCalls

phoneRingsphoneRingsunlessLoudmusicunlessLoudmusic

Unknown information :Unknown information :

JohnCalls JohnCalls Alarm AlarmJohnCalls JohnCalls PhoneRingsPhoneRingsMaryCalls MaryCalls Alarm Alarm LoudmusicLoudmusic

Alarm Alarm BurglaryBurglaryAlarm Alarm EarthQuakeEarthQuake

Observation:Observation: JohnCalls JohnCalls

abduceabduce

PhoneRingsPhoneRingsdeducededuce

AlarmAlarm

abduceabduce

BurglaryBurglary

abduceabduce

EarthQuakeEarthQuake

3 possible solutions:3 possible solutions: PhoneRings PhoneRings Burglary Burglary EarthQuake EarthQuake

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Abduction in burglar-alarm (2):Abduction in burglar-alarm (2):

Observation:Observation: JohnCalls JohnCalls MaryCallsMaryCalls LoudMusic LoudMusic

JohnCalls JohnCalls Alarm AlarmJohnCalls JohnCalls PhoneRingsPhoneRingsMaryCalls MaryCalls Alarm Alarm LoudmusicLoudmusic

Alarm Alarm BurglaryBurglaryAlarm Alarm EarthQuakeEarthQuake

abduceabduce

PhoneRingsPhoneRingsdeducededuce

AlarmAlarm deducededuce AlarmAlarm

hypothesis of hypothesis of alarmalarm is is inconsistent with observationsinconsistent with observations

1 possible explanation:1 possible explanation: PhoneRingsPhoneRings

Note: abductive procedures may be complicated !Note: abductive procedures may be complicated !

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Belief Networks orBelief Networks orBayesian NetsBayesian Nets

The The probabilistic versionprobabilistic version of the burglary-alarm example: of the burglary-alarm example:

burglaryburglary earthquakeearthquake

alarmalarm

MaryCallsMaryCallsJohnCallsJohnCalls

P(E)P(E).002.002

P(B)P(B).001.001

B E P(A)B E P(A)T T .95T T .95T F .94T F .94F T .29F T .29F F .001F F .001

A P(J)A P(J)T .90T .90F .05F .05

A P(M)A P(M)T .70T .70F .01F .01

Acyclic directed network !Acyclic directed network !

Prior probability for roots, conditional (on parents) Prior probability for roots, conditional (on parents) for lower levelsfor lower levels

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Inference in Belief NetworksInference in Belief Networksburglaryburglary earthquakeearthquake

alarmalarm

MaryCallsMaryCallsJohnCallsJohnCalls

P(E)P(E).002.002

P(B)P(B).001.001

A P(M)A P(M)T .70T .70F .01F .01

A P(J)A P(J)T .90T .90F .05F .05

B E P(A)B E P(A)T T .95T T .95T F .94T F .94F T .29F T .29F F .001F F .001

0.0160.016 (Bayes)(Bayes)

Many (all) types of questions can be answered, using Bayes Many (all) types of questions can be answered, using Bayes Rule.Rule.

What is the probability that there is What is the probability that there is no burglaryno burglary, , nor nor earthquakeearthquake, but that the , but that the alarm wentalarm went and both and both John and John and

Mary Mary calledcalled??

= P (J = P (J M M A A B B E) = 0.00062 E) = 0.00062

What is the probability that there is a What is the probability that there is a burglaryburglary, given that , given that JohnJohn callscalls??

= P(B | J) = ?= P(B | J) = ?

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An application: GENINFERAn application: GENINFER A couple is expecting a child.A couple is expecting a child.

The (expecting) mother has a The (expecting) mother has a hemophiliac risk hemophiliac risk determinedetermine the probability of hemophiliacthe probability of hemophiliac for the childfor the child

Hemophiliac disease is genetically determined:Hemophiliac disease is genetically determined: Due to a defected Due to a defected XX chromosome chromosome

mothermother fatherfather

Chromosomes:Chromosomes: XX XX X X yy

childchild

XX XX XX y yXX XX XX y y

carriercarrier hemophiliachemophiliac

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The Bayesian Network:The Bayesian Network:

mother_carriermother_carrier father_hemophfather_hemoph

child_recessivechild_recessive

P(M)P(M).00008.00008

P(F)P(F).00008.00008

M F P(C)M F P(C)T T .75T T .75T F .50T F .50F T .50F T .50F F 0F F 0

A family tree:A family tree:

okok

okok

okok

HH

??

fatherfathermothermother

grandfathergrandfather

great grandfathergreat grandfathergreat grandmothergreat grandmother

grandmothergrandmothergreat unclegreat uncle

CC

??

??

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Expanding to full network:Expanding to full network:

GGMGGM GGFGGF

GMGM GFGFGUGU

MM FF

CC

P(GGM)P(GGM) 11

P(GGF)P(GGF) 00

P(GF)P(GF) 00

P(F)P(F) 00

P(GU)P(GU) 11

Tempting solution:Tempting solution:but these are not priorbut these are not priorprobabilitiesprobabilities

But in fact remains correctBut in fact remains correctif you interpret events differentlyif you interpret events differently

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Expanding to full network (2)Expanding to full network (2)

GGMGGM GGFGGF

GMGM GFGFGUGU

MM FF

CC

P(GGM)P(GGM).00008.00008

P(GGF)P(GGF).00008.00008

P(GF)P(GF).00008.00008

P(F)P(F).00008.00008M F P(C)M F P(C)

T T .75T T .75T F .50T F .50F T .50F T .50F F 0F F 0

All dependenciesAll dependenciesare based on:are based on:

Compute: P(GGM| GU Compute: P(GGM| GU GGF) = 1 GGF) = 1 Compute: P(GM| GGM Compute: P(GM| GGM GGF) = 0.5 , etc. GGF) = 0.5 , etc.

0.50.5

0.250.25

0.1250.125

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And if there are uncles?And if there are uncles?

GGMGGM GGFGGF

GMGM GFGFGUGU

MM FF

CC

U1U1 U2U2

Recompute: P(GM| GMM Recompute: P(GM| GMM GGF GGF U1 U1 U2) U2) Propagate the information to Mother and Child Propagate the information to Mother and Child

0.50.5

0.0280.028

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And brothers?And brothers?

GGMGGM GGFGGF

GMGM GFGFGUGU

MM FF

CC

U1U1 U2U2

B1B1 B2B2 B3B3

Probability under additional condition of 3 healthy bothers: Probability under additional condition of 3 healthy bothers:

further decreasefurther decrease

0.0070.007

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Belief Networks as Rule SystemsBelief Networks as Rule Systems

Burglary Burglary (0.01)(0.01) Earthquake Earthquake (0.02)(0.02)

Alarm Alarm (0.95)(0.95) Burglary Burglary Earthquake EarthquakeAlarm Alarm (0.94)(0.94) Burglary Burglary EarthquakeEarthquakeAlarm Alarm (0.29)(0.29) Burglary Burglary Earthquake EarthquakeAlarm Alarm (0.001)(0.001) Burglary Burglary EarthquakeEarthquake

JohnCalls JohnCalls (0.90)(0.90) Alarm AlarmJohnCalls JohnCalls (0.05)(0.05) AlarmAlarm

MaryCalls MaryCalls (0.70)(0.70) Alarm AlarmMaryCalls MaryCalls (0.01)(0.01) AlarmAlarm

Doesn’t add anything …but shows that you need Doesn’t add anything …but shows that you need many rulesmany rules to represent the full Bayesian Net to represent the full Bayesian Net

In many cases you may not have all this information!In many cases you may not have all this information!

burglaryburglary earthquakeearthquake

alarmalarm

MaryCallsMaryCallsJohnCallsJohnCalls

P(E)P(E).002.002

P(B)P(B).001.001

A P(M)A P(M)T .70T .70F .01F .01

A P(J)A P(J)T .90T .90F .05F .05

B E P(A)B E P(A)T T .95T T .95T F .94T F .94F T .29F T .29F F .001F F .001

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Opinion Nets:Opinion Nets:when dependencies are unknownwhen dependencies are unknown

Will the stock (on stock market) split ?Will the stock (on stock market) split ?

Ask the opinion of 2 brokers and of 2 mystics.Ask the opinion of 2 brokers and of 2 mystics.

OROR

OROR

ANDAND

Broker 1Broker 1

Broker 2Broker 2

Mystic 1Mystic 1

Mystic 2Mystic 2

Brokers’ opinionBrokers’ opinion

Mystics’ opinionMystics’ opinion

Overall opinionOverall opinion

Stock_split Stock_split Brokers_say_split Brokers_say_split Mystics_say_split Mystics_say_splitBrokers_say_split Brokers_say_split Boker1 Boker1 Broker2 Broker2Mystics_say_split Mystics_say_split Mystic1 Mystic1 Mystic2 Mystic2

+ opinions (in probabilities) of Brokers and Mystics+ opinions (in probabilities) of Brokers and Mystics

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Opinion Nets (2)Opinion Nets (2) Problem:Problem:

We don’t know the dependencies between the opinions We don’t know the dependencies between the opinions of the brokers and/or mystics !of the brokers and/or mystics ! How to propagate their (probabilistic) opinions?How to propagate their (probabilistic) opinions?

You need upper and lower bounds!You need upper and lower bounds!

P(E)P(E)

00

11Upperbound(E)Upperbound(E)

Lowerbound(E)Lowerbound(E)

UU(E) (E) P(E) P(E)LL(E) (E) P(E) P(E)

We try to make it We try to make it LeastLeast UpperboundsUpperbounds and and GreatestGreatest LowerboundsLowerbounds

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Rules governing the propagation Rules governing the propagation of boundsof bounds

ORORAA

BBA A oror B B… … for the for the oror connective connective

UU(A) (A) UU(A (A oror B) B)

UU(B) (B) UU(A (A oror B) B)

LL(A) (A) LL(A (A oror B) - B) - UU(B) (B)

LL(B) (B) LL(A (A oror B) - B) - UU(A) (A)

UU(A (A oror B) B) UU(A) + (A) + UU(B) (B) LL(A (A oror B) B) max[ max[LL(A) , (A) , LL(B)] (B)]

… … similar for the similar for the andand connective connective

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Some example inferences:Some example inferences:AA

BB BB

BBBBAA

AA

AA

max[P(A), P(B)] max[P(A), P(B)] P(A P(A oror B) B) P(A) + P(B) P(A) + P(B)

LL(A)(A)

LL(B)(B)max(max( ,, ))

max(max( LL(A) (A) , , LL(B)(B))) is is aa lowerbound lowerbound for P(A for P(A oror B) B)

max(max( LL(A) (A) , , LL(B)(B)) ) LL(A (A oror B) B)

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Propagation of opinions:Propagation of opinions:Using forward propagation:Using forward propagation:

00

11

0.40.40.10.1

00

11

0.40.40.30.3

OROR

00

110.80.8

UU(A (A oror B) B) UU(A) + (A) + UU(B)(B)

0.30.3

LL(A (A oror B) B) max[ max[LL(A) , (A) , LL(B)] (B)]

00

11

00

11OROR

00

110.80.8

0.30.3

0.80.8

0.80.8

UU(A) (A) UU(A (A oror B) B)

UU(B) (B) UU(A (A oror B) B)

Or backward propagation:Or backward propagation:

41

The global propagation The global propagation (restricted to lowerbounds):(restricted to lowerbounds):

OROR

OROR

ANDAND

00

110.750.75

0.250.25

00

110.660.66

0.330.33

00

11

0.150.150.150.15

00

110.850.850.850.85

LL(A (A oror B) B) max[ max[LL(A) , (A) , LL(B)] (B)]

00

11

0.330.33

00

11

0.850.85

00

11

0.18 0.18 = 0.33 + = 0.33 + 0.85 -10.85 -1

LL(A (A andand B) B) LL(A) + (A) + LL(B) - 1(B) - 1

Not just OpinionsNot just Opinions: : anyany probability propagation inprobability propagation in absence absence of dependenciesof dependencies needs to apply approximations.needs to apply approximations.

42

Going Fuzzy …Going Fuzzy …for a few minutes.for a few minutes.

Examples of Fuzzy statements:Examples of Fuzzy statements: The motor is running The motor is running very hotvery hot.. Tom is a Tom is a very tallvery tall guy. guy. Electric cars are Electric cars are not very fastnot very fast.. HighHigh-performance drives require -performance drives require very rapidvery rapid dynamics dynamics and and preciseprecise regulation. regulation. Leuven is Leuven is quite a short distancequite a short distance from Brussels. from Brussels. Leuven is a Leuven is a beautifulbeautiful city. city. The maximum range of an electronic vehicle is The maximum range of an electronic vehicle is shortshort..

If If shortshort means means: 300 km or less, would 301 km be : 300 km or less, would 301 km be longlong ? ? Want to express Want to express to what degreeto what degree a property holds. a property holds.

43

Relations, sets and functionsRelations, sets and functions

Offer Offer alternative representationsalternative representations of logical statementsof logical statements

loves(John, Ann)loves(John, Ann)loves(Mary, Phil)loves(Mary, Phil)loves(Carl,Susan)loves(Carl,Susan)

Relations:Relations:

loves = loves = {(John,Mary), {(John,Mary), (Mary, Phil), (Mary, Phil), (Carl, Susan)}(Carl, Susan)}

Sets:Sets:

loves(John,Mary) = 1loves(John,Mary) = 1loves(Mary,Phil) = 1loves(Mary,Phil) = 1loves(Carl,Susan) = 1loves(Carl,Susan) = 1

Functions to {0,1}:Functions to {0,1}:

Equivalent !Equivalent !

This one allows refined statementsThis one allows refined statements

44

Fuzzy sets:Fuzzy sets: Are functions: Are functions: f:f: domaindomain [0,1][0,1]

11

00150150 160160 170170 180180 190190 200200 210210 cmcm

Crisp set Crisp set (tall men)(tall men)::

Fuzzy set Fuzzy set (tall men)(tall men)::11

00150150 160160 170170 180180 190190 200200 210210 cmcm

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Representing a domain:Representing a domain:

Crisp sets Crisp sets (men’s height)(men’s height)::

11

00150150 160160 170170 180180 190190 200200 210210 cmcm

shortshort mediummedium talltall

Fuzzy set Fuzzy set (men’s height)(men’s height)::

11

00150150 160160 170170 180180 190190 200200 210210 cmcm

shortshort mediummedium talltallshortshort

46

And much, MUCH MORE ...And much, MUCH MORE ...… in the coming lessons.… in the coming lessons.