kuliah 56 uncertainty.ppt

7/24/2019 kuliah 56 uncertainty.ppt

http://slidepdf.com/reader/full/kuliah-56-uncertaintyppt 1/46

© Negnevitsky, Pearson Education, 2005 1

Introduction, or what is uncertainty?Introduction, or what is uncertainty?

Basic probability theoryBasic probability theory

Bayesian reasoningBayesian reasoning

Bias of the Bayesian methodBias of the Bayesian method

Certainty factors theory and evidentialCertainty factors theory and evidentialreasoning reasoning

SummarySummary

Lecture 3Lecture 3

Uncertainty management in ruleUncertainty management in rule

based e!pert systemsbased e!pert systems




Information can be incomplete, inconsistent,Information can be incomplete, inconsistent,uncertain, or all three. In other words, informationuncertain, or all three. In other words, information

is often unsuitable for solving a problem.is often unsuitable for solving a problem.

UncertaintyUncertainty is defined as the lack of the exactis defined as the lack of the exact

knowledge that would enable us to reach a perfectlyknowledge that would enable us to reach a perfectly

reliable conclusion. Classical logic permits onlyreliable conclusion. Classical logic permits only

exact reasoning. It assumes that perfect knowledgeexact reasoning. It assumes that perfect knowledge

always exists and thealways exists and the law of the excluded middlelaw of the excluded middle

can always be applied:can always be applied:

I"I" A A is trueis true I"I" A A is falseis false

#$%&#$%& A A is not falseis not false #$%&#$%& A A is not trueis not true

Introduction, or what is uncertainty?Introduction, or what is uncertainty?




'ea( implications'ea( implications . . Domain experts andDomain experts andknowledge engineers have the painful taskknowledge engineers have the painful task

of establishing concrete correlations between IFof establishing concrete correlations between IF

condition! and "#$% action! parts of the rules.condition! and "#$% action! parts of the rules.

"herefore, expert systems need to have the ability"herefore, expert systems need to have the ability

to handle vague associations, for example byto handle vague associations, for example by

accepting the degree of correlations as numericalaccepting the degree of correlations as numerical

certainty factors.certainty factors.

Sources of uncertain (nowledgeSources of uncertain (nowledge




Imprecise language)Imprecise language) &ur natural language is&ur natural language is

ambiguous and imprecise. 'e describe factsambiguous and imprecise. 'e describe facts

with such terms aswith such terms as oftenoften andand

sometimes sometimes,, frequently frequently andand hardly ever hardly ever ..

(s a result, it can be difficult to(s a result, it can be difficult toexpress knowledge in the precise IF)"#$% form ofexpress knowledge in the precise IF)"#$% form of

production rules. #owever, if the meaning of production rules. #owever, if the meaning of

the facts is *uantified, it can bethe facts is *uantified, it can be

used in expert systems. In +--, ayused in expert systems. In +--, ay

/impson asked 011 high school and/impson asked 011 high school and

college students to place 23 terms likecollege students to place 23 terms like oftenoften on aon a

scale between + and +33. In +45, 6iltonscale between + and +33. In +45, 6ilton




*uantification of ambiguous and imprecise*uantification of ambiguous and imprecise

terms on a timefre+uency scaleterms on a timefre+uency scale

Term

(lways7ery often8sually&ften9enerallyFre*uentlyather often(bout as often as not

%ow and then/ometimes&ccasionally&nce in a while

%ot often8sually not/eldom#ardly ever 7ery seldomarely(lmost never

%ever

Mean value

55515504113232323+1

+0+3+34103

Term

(lways7ery often8sually&ften

9enerallyFre*uentlyather often

(bout as often as not %ow and then/ometimes&ccasionally&nce in a while

%ot often8sually not/eldom#ardly ever 7ery seldomarely(lmost never

%ever

Mean value

+335--22130-22522

+4+45123

Milton Hakel +45! Ray Simpson +--!




Un(nown dataUn(nown data . . 'hen the data is incomplete or'hen the data is incomplete or

missing, the only solution is to accept themissing, the only solution is to accept thevalue ;unknown< and proceed to anvalue ;unknown< and proceed to anapproximateapproximate reasoning with thisreasoning with thisvalue.value.

Combining the views of different e!pertsCombining the views of different e!perts . . =arge=argeexpert systems usually combine the knowledge andexpert systems usually combine the knowledge andexpertise of a number of experts. 8nfortunately,expertise of a number of experts. 8nfortunately,

experts often have contradictory opinions andexperts often have contradictory opinions and produce conflicting rules. "o resolve the conflict, produce conflicting rules. "o resolve the conflict,the knowledge engineer has to attach a weight tothe knowledge engineer has to attach a weight toeach expert and then calculate the compositeeach expert and then calculate the composite

conclusion. >ut no systematic method exists toconclusion. >ut no systematic method exists toobtain these wei hts.obtain these wei hts.


http://slidepdf.com/reader/full/kuliah-56-uncertaintyppt 7/46 © Negnevitsky, Pearson Education, 2005 7

"he concept of probability has a long history that"he concept of probability has a long history that goes back thousands of years when words likegoes back thousands of years when words like

;probably<, ;likely<, ;maybe<, ;perhaps< and;probably<, ;likely<, ;maybe<, ;perhaps< and

;possibly< were introduced into spoken languages.;possibly< were introduced into spoken languages.

#owever, the mathematical theory of probability#owever, the mathematical theory of probability

was formulated only in the +th century.was formulated only in the +th century.

"he"he probabilityprobability of an event is the proportion ofof an event is the proportion of

cases in which the event occurs. ?robability cancases in which the event occurs. ?robability can

also be defined as aalso be defined as a scientific measure of chancescientific measure of chance..

Basic probability theoryBasic probability theory



?robability can be expressed mathematically as a?robability can be expressed mathematically as a

numerical index with a range betweennumerical index with a range between

@ero an absolute impossibility! to@ero an absolute impossibility! tounity an absoluteunity an absolute certainty!. certainty!.

6ost events have a probability index strictly6ost events have a probability index strictly

between 3 and +, which means that each event has between 3 and +, which means that each event has

at leastat least two possible outcomes: favourable outcometwo possible outcomes: favourable outcome

or success, and unfavourable outcome or failure. or success, and unfavourable outcome or failure.

the number of possible outcomes

the number of successes

success! P =

the number of possible outcomes failure! P =

the number of failures



IfIf s s is the number of times success can occur, andis the number of times success can occur, and f f

is the number of times failure can occur, thenis the number of times failure can occur, then

If we throw a coin, the probability of getting a headIf we throw a coin, the probability of getting a headwill be e*ual to the probability of getting a tail. In awill be e*ual to the probability of getting a tail. In a

single throw,single throw, s s AA f f A+, and therefore the probabilityA+, and therefore the probability

of getting a head or a tail! is 3.1.of getting a head or a tail! is 3.1.

aa

ndnd

f ss psuccessP+

==

f s

f

qfailureP +==

p+ q= +



=et=et A A be an event in the world and be an event in the world and B B be another event. be another event.

/uppose that events/uppose that events A A andand B B are not mutuallyare not mutuallyexclusive, but occur conditionally on theexclusive, but occur conditionally on the

occurrence of the other. "he probability that eventoccurrence of the other. "he probability that event

A A will occur if eventwill occur if event B B occurs is called theoccurs is called the

conditional probability)conditional probability) ConditionalConditional

probability is denoted mathematically as probability is denoted mathematically as p p A A|| B B!! inin

which the vertical bar representswhich the vertical bar represents G!"# G!"# and theand the

complete probability expression is interpreted ascomplete probability expression is interpreted as$%onditional probability of event A occurrin& &iven$%onditional probability of event A occurrin& &iven

that event B has occurred' that event B has occurred'..

Conditional probabilityConditional probability

occur canBtimesof number the

occur canBand Atimesof number theB A p =



"he number of times"he number of times A A andand B B can occur, or thecan occur, or the

probability that both probability that both A A andand B B will occur, is calledwill occur, is called

thethe oint probability oint probability ofof A A andand B B. It is represented. It is representedmathematically asmathematically as p p A A∩∩ B B!!. "he number of ways. "he number of ways B B

can occur is the probability ofcan occur is the probability of B B,, p p B B!, and thus!, and thus

/imilarly, the conditional probability of event/imilarly, the conditional probability of event B B

occurring given that eventoccurring given that event A A has occurred e*ualshas occurred e*uals

B pB A pB A p ∩=

A p

AB p AB p

∩=



#ence,#ence,or or

/ubstituting the last e*uation into the e*uation/ubstituting the last e*uation into the e*uation

yields theyields the Bayesian ruleBayesian rule::

A p AB p AB p ×=∩

A p AB pB A p ×=∩

B p

B A pB A p

∩=



where:where:

p p A A|| B B! is the conditional probability that event! is the conditional probability that event A Aoccurs given that eventoccurs given that event B B has occurredBhas occurredB

p p B B|| A A! is the conditional probability of event! is the conditional probability of event B B

occurring given that eventoccurring given that event A A has occurredBhas occurredB

p p A A! is the probability of event! is the probability of event A A occurringBoccurringB p p B B! is the probability of event! is the probability of event B B occurring.occurring.

BayesianBayesian rulerule

B p A p AB pB A p ×=



#he oint probability#he oint probability

i

n

ii

n

ii B pB A pB A p ×=∩ ∑∑ == 11

AB4

B3

B1

B2



If the occurrence of eventIf the occurrence of event A A depends on only twodepends on only two

mutually exclusive events,mutually exclusive events, B B and %&"and %&" B B , we obtain:, we obtain:

wherewhere ¬¬ is the logical function %&".is the logical function %&".

/imilarly,/imilarly,

/ubstituting this e*uation into the >ayesian rule yields:/ubstituting this e*uation into the >ayesian rule yields:

p( A)= p( A|B) p(B)+ p( A¬B) × p(¬B)

p(B)= p(B A) × p( A)+ p(B|¬ A) × p(¬ A)

A p AB p A p AB p

A p AB pB A p

¬×¬+×

×=




/uppose all rules in the knowledge base are/uppose all rules in the knowledge base arerepresented in the following form:represented in the following form:

"his rule implies that if event"his rule implies that if event " " occurs, then theoccurs, then the

probability that event probability that event H H will occur iswill occur is p p..

In expert systems,In expert systems, H H usually represents a hypothesisusually represents a hypothesis

andand " " denotes evidence to support this hypothesis.denotes evidence to support this hypothesis.

Bayesian reasoningBayesian reasoning

IFIF " " is trueis true

"#$%"#$% H H is true with probabilityis true with probability p p




"he >ayesian rule expressed in terms of hypotheses"he >ayesian rule expressed in terms of hypotheses

and evidence looks like this:and evidence looks like this:

where:where: p p H H ! is the prior probability of hypothesis! is the prior probability of hypothesis H H being trueB being trueB

p p " " || H H ! is the probability that hypothesis! is the probability that hypothesis H H being true will being true will

result in evidenceresult in evidence " " BB

p p¬¬ H H ! is the prior probability of hypothesis! is the prior probability of hypothesis H H being being

falseBfalseB

p p " " |¬|¬ H H ! is the probability of finding evidence! is the probability of finding evidence " " eveneven

when hypothesiswhen hypothesis H H is false.is false.

H pHE pH pHE p

H pHE pEH p

¬×¬+×

×=




In expert systems, the probabilities re*uired to solveIn expert systems, the probabilities re*uired to solve

a problem are provided by experts. (n experta problem are provided by experts. (n expert

determines thedetermines the prior probabilitiesprior probabilities for possiblefor possiblehypotheseshypotheses p p H H ! and! and p p¬¬ H H !, and also the!, and also the

conditional probabilitiesconditional probabilities for observing evidencefor observing evidence " "

if hypothesisif hypothesis H H is true,is true, p p " " || H H !, and if hypothesis!, and if hypothesis H H

is false,is false, p p " " |¬|¬ H H !.!.

8sers provide information about the evidence8sers provide information about the evidence

observed and the expert system computesobserved and the expert system computes p p H H || " " ! for! for

hypothesishypothesis H H in light of the user)supplied evidencein light of the user)supplied evidence " " . ?robability. ?robability p p H H || " " ! is called the! is called the posteriorposterior

probabilityprobability of hypothesisof hypothesis H H upon observingupon observing

evidenceevidence " " ..




'e can take into account both multiple hypotheses'e can take into account both multiple hypotheses

H H ++,, H H

22,...,,..., H H mm and multiple evidencesand multiple evidences " "

++,, " " 22 ,...,,..., " "

nn..

"he hypotheses as well as the evidences must be"he hypotheses as well as the evidences must bemutually exclusive and exhaustive.mutually exclusive and exhaustive.

/ingle evidence/ingle evidence " " and multiple hypotheses follow:and multiple hypotheses follow:

6ultiple evidences and multiple hypotheses6ultiple evidences and multiple hypotheses follow:follow:

∑=

×

×= m

k k k

iii

H p H " p

H p H " p " H p

1

∑=

×

×=

m

k k k n

iinni

H p H " ((( " " p

H p H " ((( " " p " ((( " " H p

121

2121




"his re*uires to obtain the conditional probabilities"his re*uires to obtain the conditional probabilities

of all possible combinations ofof all possible combinations ofevidences for all hypotheses, and thus places anevidences for all hypotheses, and thus places an

enormous burden on the expert.enormous burden on the expert.

"herefore, in expert systems, conditional"herefore, in expert systems, conditionalindependence among different evidences assumed.independence among different evidences assumed.

"hus, instead of the unworkable"hus, instead of the unworkable equationequation, we, we

attain: attain:

∑=

××××

××××m

k k k nk k

iiniini

H p H " p((( H " p H " p

H p H " p H " p H " p " ((( " " H p

121

2121

...=




=et us consider a simple example.=et us consider a simple example.

/uppose an expert, given three conditionally/uppose an expert, given three conditionally

independent evidencesindependent evidences " " ++,, " "

22 andand " " 00, creates three, creates three

mutually exclusive and exhaustive hypothesesmutually exclusive and exhaustive hypotheses H H ++,, H H 22 andand H H

00, and provides prior probabilities for these, and provides prior probabilities for these

hypotheses Ehypotheses E p p H H ++!,!, p p H H

22! and! and p p H H 00!, respectively.!, respectively.

"he expert also determines the conditional"he expert also determines the conditional probabilities of observing each evidence for all probabilities of observing each evidence for all

possible hypotheses. possible hypotheses.

-an(ing potentially true hypotheses-an(ing potentially true hypotheses




#he prior and conditional probabilities#he prior and conditional probabilities

(ssume that we first observe evidence(ssume that we first observe evidence " " 00. "he expert. "he expert

system computes the posterior probabilities for allsystem computes the posterior probabilities for all

hypotheses ashypotheses as

H y po t h es i sProbability

= 1i = 2i = 3i

0.40

0.9

0.6

0.3

0.35

0.0

0.7

0.8

0.25

0.7

0.9

0.5

iH p

iHE p 1

iHE p 2

iHE p 3




(fter evidence(fter evidence " " 00 is observed, belief in hypothesisis observed, belief in hypothesis H H ++

decreases and becomes e*ual to belief in hypothesisdecreases and becomes e*ual to belief in hypothesis

H H 22. >elief in hypothesis. >elief in hypothesis H H

00 increases and even nearlyincreases and even nearly

reaches beliefs in hypothesesreaches beliefs in hypotheses H H ++ andand H H 22..

"hus,"hus,

02,+,A,0

+

0

00 i

H p H " p

H p H " p " H p

k

k k

iii

∑=

×

×=

3.0-21.3.301.3.33.-33.4

3.-33.40+ =

×××

×= " H p

3.0-21.3.301.3.33.-33.4

01.3.302 =×××

×= " H p

3.0221.3.301.3.33.-33.4

21.3.300 =

×××

×= " H p




/uppose now that we observe evidence/uppose now that we observe evidence " " ++. "he. "he

posterior probabilities are calculated as posterior probabilities are calculated as

#ence,#ence,

32,1,=,3

131

3131 i

H pHE pHE p

H pHE pHE pEEH p

k k k k

iiii

∑=

××

××=

0.1925.00.5+35.07.00.8+0.400.60.3

0.400.60.3311 =

××××××

××=EEH p

0.5225.00.5+35.07.00.8+0.400.60.3

35.07.00.8

312

=××××××

××=EEH p

0.2925.00.5+35.07.00.8+0.400.60.3

25.09.00.5313 =

××××××××

=EEH p

#ypothesis#ypothesis H H 22 has now become the most likely one.has now become the most likely one.




(fter observing evidence(fter observing evidence " " 22, the final posterior, the final posterior

probabilities for all hypotheses are calculated: probabilities for all hypotheses are calculated:

(lthough the initial ranking was(lthough the initial ranking was H H ++,, H H

22 andand H H 00, only, only

hypotheseshypotheses H H ++ andand H H

00 remain under considerationremain under consideration afterafter

all evidences all evidences " " ++,, " " 22 andand " " 00! were observed.! were observed.

32,1,=,3

1321

321321 i

H pHE pHE pHE p

H pHE pHE pHE pEEEH p

k k k k k

iiiii

∑=

×××

×××=

0.4525.09.00.50.7

0.7

0.7

0.5

+.3507.00.00.8+0.400.60.90.3

0.400.60.90.3

3211=

×××××××××

×××=EEEH p

025.09.0+.3507.00.00.8+0.400.60.90.3

35.07.00.00.83212 =

×××××××××

×××=EEEH p

0.55

25.09.00.5+.3507.00.00.8+0.400.60.90.3

25.09.00.70.53213 =

×××××××××

×××=EEEH p




"he framework for >ayesian reasoning re*uires"he framework for >ayesian reasoning re*uires probability values as primary inputs. "he probability values as primary inputs. "he

assessment of these values usually involves humanassessment of these values usually involves human

Gudgement. #owever, psychological research Gudgement. #owever, psychological research

shows that humans cannot elicit probability valuesshows that humans cannot elicit probability values

consistent with the >ayesian rules.consistent with the >ayesian rules.

"his suggests that the conditional probabilities may"his suggests that the conditional probabilities may

be inconsistent with the prior probabilities be inconsistent with the prior probabilitiesgiven bygiven by the expert.the expert.

Bias of the Bayesian methodBias of the Bayesian method




Consider, for example, a car that does not start andConsider, for example, a car that does not start and

makes odd noises when you press the starter. "hemakes odd noises when you press the starter. "he conditional probability of the starter being faulty if conditional probability of the starter being faulty if the car makes odd noises may be expressedthe car makes odd noises may be expressed

as:as:

IFIF the symptom is ;odd noises<the symptom is ;odd noises<"#$% the starter is"#$% the starter is

bad with probability 3. bad with probability 3.

Consider, for example, a car that does not start andConsider, for example, a car that does not start andmakes odd noises when youmakes odd noises when you press the starter. "he conditional press the starter. "he conditional probability of the starter being faulty if probability of the starter being faulty if

the car makes odd noises may bethe car makes odd noises may beex ressed as:ex ressed as:




"herefore, we can obtain a companion rule that states"herefore, we can obtain a companion rule that states

IFIF the symptom is ;odd noises<the symptom is ;odd noises<

"#$% the starter is good with probability 3.0"#$% the starter is good with probability 3.0 Domain experts do not deal with conditionalDomain experts do not deal with conditional

probabilities and often deny the very existence of the probabilities and often deny the very existence of the hidden implicit probabilityhidden implicit probability 3.0 in our example!.3.0 in our example!.

'e would also use available statistical information'e would also use available statistical informationand empirical studies to derive the following rules:and empirical studies to derive the following rules:

IFIF the starter is badthe starter is bad

"#$% the symptom is ;odd noises< probability 3.51"#$% the symptom is ;odd noises< probability 3.51

IFIF the starter is badthe starter is bad"#$% the symptom is not ;odd noises< probability 3.+1"#$% the symptom is not ;odd noises< probability 3.+1




"o use the >ayesian rule, we still need the"o use the >ayesian rule, we still need the priorprior

probabilityprobability, the probability that the starter is bad if, the probability that the starter is bad if

the car does not start. /uppose, the expert suppliesthe car does not start. /uppose, the expert supplies

us the value of 1 per cent. %ow we can apply theus the value of 1 per cent. %ow we can apply the

>ayesian rule to obtain:>ayesian rule to obtain:

#he number obtained is significantly lower#he number obtained is significantly lower

than the e!pert.s estimate of /)0 given at thethan the e!pert.s estimate of /)0 given at the

beginning of this section)beginning of this section)

"he reason for the inconsistency is that the expert"he reason for the inconsistency is that the expert

made different assumptions when assessing themade different assumptions when assessing the

conditional and prior probabilities. conditional and prior probabilities.

3.20

3.13.+13.313.51

3.313.51=

××

×=noisesodd bad is starter p




Certainty factors theory is a popular alternative toCertainty factors theory is a popular alternative to

>ayesian reasoning.>ayesian reasoning.

(( certainty factorcertainty factor cfcf !, a number to measure the!, a number to measure the

expertHs belief. "he maximum value of the certaintyexpertHs belief. "he maximum value of the certainty

factor is, say, +.3 definitely true! and thefactor is, say, +.3 definitely true! and the

minimumminimum −−+.3 definitely false!. For example,+.3 definitely false!. For example, ifif

the expert states that some evidence is almostthe expert states that some evidence is almostcertainly true, acertainly true, a cfcf value of 3.5 would be assignedvalue of 3.5 would be assigned

to this evidence.to this evidence.

Certainty factors theory andCertainty factors theory and

evidential reasoningevidential reasoning




Uncertain terms and theirUncertain terms and their

interpretation in 12CI& interpretation in 12CI&

Term

Definitely not(lmost certainly not

?robably not6aybe not8nknown

%ertainty )actor

3.-3.43.5+.3

6aybe?robably(lmost certainlyDefinitely

+.3

3.5

3.4

3.-

3.2 to 3.2




In expert systems with certainty factors, theIn expert systems with certainty factors, the

knowledge base consists of a set of rules that haveknowledge base consists of a set of rules that havethe following syntax:the following syntax:

IFIF

<<evidenceevidence>>"#$%"#$% <<hypothesishypothesis>> cfcf

wherewhere cfcf represents belief in hypothesisrepresents belief in hypothesis H H givengiven

that evidencethat evidence " " has occurred.has occurred.




"he certainty factors theory is based on two functions:"he certainty factors theory is based on two functions:

measure of beliefmeasure of belief MB MB H H ,, " " !, and measure of disbelief!, and measure of disbelief

M* M* H H ,, " " !.!.

p p H H ! is the prior probability of hypothesis! is the prior probability of hypothesis H H being trueB being trueB

p p H H || " " ! is the probability that hypothesis! is the probability that hypothesis H H is true givenis true given

evidenceevidence " " ..

if p H ! A +

MB H , " ! A

+

ma+ [+, 3] − p H !

ma+ [ p H | " !, p H !] −

−

p H !otherwise

if p H ! A 3

M* H , " ! A

+

min [+, 3] − p H !

min [ p H | " !, p H !] p H !otherwise




"he values of"he values of MB MB H H ,, " " ! and! and M* M* H H ,, " " ! range! range

between 3 and +. "he strength of belief or between 3 and +. "he strength of belief or

disbelief in hypothesisdisbelief in hypothesis H H depends on the kind of depends on the kind of evidenceevidence " " observed. /ome facts may increase theobserved. /ome facts may increase the

strength of belief, but some increase the strength of strength of belief, but some increase the strength of

disbelief.disbelief.

#he total strength of belief or disbelief in a#he total strength of belief or disbelief in a

hypothesishypothesis

EH,MD,EH,MBmin- EH,MDEH,MB=cf 1 −




%!ample%!ample

Consider a simple rule:Consider a simple rule:

IFIF A A isis , ,

"#$%"#$% B B isis --

(n expert may not be absolutely certain that this rule(n expert may not be absolutely certain that this rule

holds. (lso suppose it has been observed thatholds. (lso suppose it has been observed that

in some cases, even when the IF part of thein some cases, even when the IF part of the

rule is satisfied andrule is satisfied and ob.ect Aob.ect A takes ontakes on value , value , , obGect, obGect B B

can ac*uire somecan ac*uire some different valuedifferent value / / ..IFIF A A isis , ,

"#$%"#$% B B isis -- cfcf 3.B3.B

B B isis / / cfcf 3.23.2




"he certainty factor assigned by a rule is propagated"he certainty factor assigned by a rule is propagated

through the reasoning chain. "his involvesthrough the reasoning chain. "his involves

establishing the net certainty of the rule conse*uentestablishing the net certainty of the rule conse*uent

when the evidence in the rule antecedent is uncertain:when the evidence in the rule antecedent is uncertain:

cfcf 44 H H ,, E E 5 65 6 cfcf 44 E E 5 !5 ! cfcf

For example,For example,IFIF sky is clear sky is clear

"#$% the forecast is sunny "#$% the forecast is sunny cfcf

3.53.5

and the current certainty factor ofand the current certainty factor of sky is clear sky is clear is 3.1is 3.1,,

thenthen

AA ⋅⋅ AA




For conGunctive rules such asFor conGunctive rules such as

the certainty of hypothesisthe certainty of hypothesis H H , is established as follows:, is established as follows:

cfcf 44 H H ,, E E 77∩ E E

88∩99∩

%%nn5 65 6 minmin ::cfcf 44 E E 775,5, cfcf 44 E E

885,))),5,))),cfcf 44 E E nn5;5; cfcf

For example,For example,

IFIF sky is clearsky is clear

(%D the forecast is sunny(%D the forecast is sunny

"#$% the action is Jwear sunglassesH "#$% the action is Jwear sunglassesH cfcf 3.53.5 and the certainty ofand the certainty of sky is clear sky is clear is 3.is 3. and the certainty of theand the certainty of the

forecast of sunny forecast of sunny is 3.is 3., then, then

cfcf H H ,, " " ++∩∩ " "

22! A! A minmin K3., 3.LK3., 3.L ⋅⋅ 3.5 A 3.3.5 A 3. ⋅⋅ 3.5 A 3.143.5 A 3.14

IF <evidenceE1>.

.

.

AND <evidenceEn> THEN <!"#$e%i%H> &cf '




For disGunctive rules such asFor disGunctive rules such as

the certainty of hypothesisthe certainty of hypothesis H H , is established as follows:, is established as follows:

cfcf 44 H H ,, E E 77∪ E E

88∪99∪

%%nn5 65 6 maxmax ::cfcf 44 E E 775,5, cfcf 44 E E

885,))),5,))),cfcf 44 E E nn5;5; cf cf

For example,For example,

IFIF sky is overcastsky is overcast

&& the forecast is rainthe forecast is rain

"#$% the action is Jtake an"#$% the action is Jtake an

umbrellaH umbrellaH cfcf 3.3.

and the certainty ofand the certainty of sky is overcast sky is overcast is 3.4is 3.4 and the certainty of theand the certainty of the

forecast of rain forecast of rain is 3.5is 3.5, then, then

cfcf H H ,, " " 77∪∪ " "

88 ! A! A ma+ma+ K3.4, 3.5LK3.4, 3.5L ⋅⋅ 3. A 3.53. A 3.5 ⋅⋅ 3. A 3.23. A 3.2

IF <evidenceE1>.

.

.

<evidenceEn> THEN <!"#$e%i%H> &cf '




'hen the same conse*uent is obtained as a result of'hen the same conse*uent is obtained as a result ofthe execution of two or more rules, the individualthe execution of two or more rules, the individualcertainty factors of these rules must be merged tocertainty factors of these rules must be merged to

give a combined certainty factor for a hypothesis.give a combined certainty factor for a hypothesis.

/uppose the knowledge base consists of the following/uppose the knowledge base consists of the following rules:rules:

Rule Rule

+:+:

IFIF

A A

isis

, ,

"#$%"#$% %% isis / / cfcf 3.53.5

Rule Rule 2:2: IFIF B B isis - - "#$%"#$% %% isis / / cfcf 3.43.4

'hat certainty should be assigned to obect'hat certainty should be assigned to obect CC

having valuehaving value Z Z if bothif both Rule Rule 7 and7 and Rule Rule 8 are8 are

fired?fired?




Common sense suggests that, if we have twoCommon sense suggests that, if we have two

pieces of evidence pieces of evidence A A isis , , andand B B isis - - ! from! from different sources different sources Rule Rule + and+ and Rule Rule 2! supporting2! supporting

the same hypothesis the same hypothesis %% isis / / !, then the confidence!, then the confidence

in this hypothesis should increase and becomein this hypothesis should increase and become

stronger than if only one piece of evidence hadstronger than if only one piece of evidence had been obtained. been obtained.




"o calculate a combined certainty factor we can"o calculate a combined certainty factor we can

use the following e*uation:use the following e*uation:

where:where:

cf cf ++ is the confidence in hypothesisis the confidence in hypothesis H H established byestablished by Rule Rule +B+Bcf cf

22 isis

the confidence in hypothesisthe confidence in hypothesis H H established byestablished by Rule Rule 2B2B

||cf cf ++|| andand ||cf cf

22|| are absolute magnitudes ofare absolute magnitudes of cf cf ++ andand cf cf

22,,

res ectivel .res ectivel .

i* cf 0 ndcf 0

cf 1+ cf 2× (1− cf 1)

cf (cf 1, cf 2) =1− min[|cf

1

|, |cf 2

|]cf 1+ cf 2

cf 1+ cf 2× (1+ cf 1)

i* cf 0 #cf 0

i* cf < 0 ndcf < 0




"he certainty factors theory provides a"he certainty factors theory provides a practical practicalalternative to >ayesian reasoning. "he heuristicalternative to >ayesian reasoning. "he heuristic

manner of combining certainty factors is differentmanner of combining certainty factors is different

from the manner in which they would be combinedfrom the manner in which they would be combined

if they were probabilities. "he certainty theory isif they were probabilities. "he certainty theory isnot ;mathematically pure< but does mimic thenot ;mathematically pure< but does mimic the

thinking process of a human expert.thinking process of a human expert.




Comparison of Bayesian reasoningComparison of Bayesian reasoning

and certainty factorsand certainty factors ?robability theory is the oldest and best)established?robability theory is the oldest and best)established

techni*ue to deal with inexact knowledge andtechni*ue to deal with inexact knowledge and

random data. It works well in such areas asrandom data. It works well in such areas as

forecasting and planning, where statistical data isforecasting and planning, where statistical data isusually available and accurate probability statementsusually available and accurate probability statements

can be made.can be made.




#owever, in many areas of possible applications of#owever, in many areas of possible applications of

expert systems, reliable statistical information is notexpert systems, reliable statistical information is not

available or we cannot assume the conditionalavailable or we cannot assume the conditional

independence of evidence. (s a result, manyindependence of evidence. (s a result, many

researchers have found the >ayesian methodresearchers have found the >ayesian method

unsuitable for their work. "his dissatisfactionunsuitable for their work. "his dissatisfaction

motivated the development of the certainty factorsmotivated the development of the certainty factorstheory.theory.

(lthough the certainty factors approach lacks the(lthough the certainty factors approach lacks the

mathematical correctness of the probability theory, itmathematical correctness of the probability theory, itoutperforms subGective >ayesian reasoning in suchoutperforms subGective >ayesian reasoning in such

areas as diagnostics.areas as diagnostics.




Certainty factors are used in cases where theCertainty factors are used in cases where the

probabilities are not known or are too difficult or probabilities are not known or are too difficult or

expensive to obtain. "he evidential reasoningexpensive to obtain. "he evidential reasoning

mechanism can manage incrementally ac*uiredmechanism can manage incrementally ac*uired

evidence, the conGunction and disGunction ofevidence, the conGunction and disGunction of

hypotheses, as well as evidences with differenthypotheses, as well as evidences with different

degrees of belief.degrees of belief. "he certainty factors approach also provides better"he certainty factors approach also provides better

explanations of the control flow through a rule)explanations of the control flow through a rule)

based expert system. based expert system.



"he >ayesian method is likely to be the most"he >ayesian method is likely to be the most

appropriate if reliable statistical data exists, theappropriate if reliable statistical data exists, the

knowledge engineer is able to lead, and the expert isknowledge engineer is able to lead, and the expert is

available for serious decision)analyticalavailable for serious decision)analytical

conversations.conversations.

In the absence of any of the specified conditions,In the absence of any of the specified conditions,

the >ayesian approach might be too arbitrary andthe >ayesian approach might be too arbitrary and

even biased to produce meaningful results.even biased to produce meaningful results.

"he >ayesian belief propagation is of exponential"he >ayesian belief propagation is of exponential

complexity, and thus is impractical for largecomplexity, and thus is impractical for largeknowledge bases.knowledge bases.

kuliah 56 uncertainty.ppt

Documents

Transcript of kuliah 56 uncertainty.ppt