cs511 uncertainty

8/12/2019 Cs511 Uncertainty

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Reasoning Under Uncertainty

Most tasks requiring intelligent behavior have some

degree of uncertainty associated with them.

The type of uncertainty that can occur in knowledge-

based systems may be caused by problems with thedata. For example:

. !ata might be missing or unavailable

". !ata might be present but unreliable orambiguous due to measurement errors.

#. The representation of the data may be imprecise

or inconsistent.

$. !ata may %ust be user&s best guess.

'. !ata may be based on defaults and the defaultsmay have exceptions.


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The uncertainty may also be caused by the

represented knowledge since it might

. (epresent best guesses of the experts that arebased on plausible or statistical associations theyhave observed.

". )ot be appropriate in all situations *e.g.+ mayhave indeterminate applicability,

iven these numerous sources of errors+ most

knowledge-based systems require the incorporation ofsome form of uncertainty management.

hen implementing some uncertainty scheme we

must be concerned with three issues:

. /ow to represent uncertain data

". /ow to combine two or more pieces of uncertaindata

#. /ow to draw inference using uncertain data

e will introduce three ways of handling uncertainty:

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0robabilistic reasoning.

1ertainty factors

!empster-2hafer Theory

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1. Classical Probability

The oldest and best defined technique for managinguncertainty is based on classical probability theory. 3et usstart to review it by introducing some terms.

Sample space: 1onsider an experiment whose

outcome is not predictable with certainty in advance./owever+ although the outcome of the experiment

will not be known in advance+ let us suppose that theset of all possible outcomes is known. This set of all

possible outcomes of an experiment is known as thesample spaceof the experiment and denoted by S.

For example:

4f the outcome of an experiment consists in the

determination of the sex of a newborn child+ then2 5 6g+ b7

where the outcomegmeans that the child is a girland bthat it is a boy.

4f the experiment consists of flipping two coins+

then the sample space consists of the followingfour points:

2 5 6*/+ /,+ */+ T,+ *T+ /,+ *T+ T,7

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Event: any subset Eof the sample space is known as

anevent.

That is+ an event is a set consisting of possibleoutcomes of the experiment. 4f the outcome of theexperiment is contained in 8+ then we say that 8 hasoccurred.

For example+ if 8 5 6*/+ /,+ 6/+ T,7+ then 8 is theevent that a head appears on the first coin.

For any event 8 we define the new event 8&+ referred

to as the complement of 8+ to consist of all points inthe sample space 2 that are not in 8.

Mutually exclusive events: 9 set of events 8+ 8"+ ...+

8nin a sample space 2+ are called mutually exclusiveevents if 8i8%5 + i %+ i+ % n.

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9 formal theory of probability can be made using

three axioms:

. 0*8, .

". 0*8i, 5 *or 0*2, 5 ,i

This axiom states that the sum of all events which donot affect each other+ called mutually exclusive

events+ is .

9s a corollary of this axiom:

0*8i, ; 0*8i&, 5 +

where 8i& is the complement of event 8i.

#. 0*88", 5 0*8, ; 0*8",+

where 8 and 8" are mutually exclusive events. 4ngeneral+ this is also true.


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Compound probabilities

8vents that do not affect each other in any way are

called independent events. For two independentevents 9 and =+

0*9 =, 5 0*9, 0*=,

Independent events: The events 8+ 8"+ ...+ 8n in asample space 2+ are independent if

0*8i... 8ik, 5 0*8i, ...0*8ik,

for each subset 6i+ ...+ik, 6+ ...+ n7+k n+ n .

4f events 9 and = are mutually exclusive+ then

0*9 =, 5 0*9, ; 0*=,

4f events 9 and = are not mutually exclusive+ then

0*9 =, 5 0*9, ; 0*=, - 0*9 =,

This is also called Addition la.

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Conditional Probabilities

The probability of an event 9+ given = occurred+ is calleda conditional probability and indicated by

0*9 ? =,

The conditional probability is defined as

0*9 =,

0*9 ? =, 5 ------------------+ for 0*=, .

0*=,

Multiplicative !aof probability for two events is

then defined as

0*9 =, 5 0*9 ? =, 0*=,

which is equivalent to the following

0*9 =, 5 0*= ? 9, 0*9,

"enerali#ed Multiplicative !a0*99"... 9n, 5

0*9? 9"... 9n, 0*9"? 9#... 9n,

... 0*9n-? 9n, 0*9n,

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An example

9s an example of probabilities+ Table below showshypothetical probabilities of a disk crash using a =rand Adrive within one year.

=rand A =rand A& Total of (ows

1rash 1 .< . .>)o crash 1& ." . .#Total of columns .@ ." .

$ypot%etical probabilities o& a dis' cras%

A A& Total of rows

1 0*1 A, 0*1 A&, 0*1,

1& 0*1&A, 0*1& A&, 0*1&,

Total of columns 0*A, 0*A&,

Probability interpretation o& to setsBsing above tables+ the probabilities of all events can becalculated. 2ome probabilities are*, 0*1, 5 .>*", 0*1&, 5 .#

*#, 0*A, 5 .@*$, 0*A&, 5 ."*', 0*1 A, 5 .'0*8 ? /&, 5 0*(ob was observed sneeing ?(ob does not have a cold,

5 ."Then

0*8, 5 0*(ob was observed sneeing,5 *.>',*.", ; *.",*.@,5 .#

and

0*/ ? 8, 50*(ob has a cold ? (ob was observed sneeing,

*.>',*.",5 ---------------

*.#,5 .$@#@>

Er (ob&s probability of having a cold given that he

sneees is about .'.

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e can also determine what his probability of having

a cold would be if he was not sneeing:

0*8& ? /,0*/,0*/ ? 8&, 5 -------------------

0*8&,*-.>', *.",

5 -------------------* - .#,

5 .>"$, that theidentity of the organism is streptococcus

This can be written in terms of posterior probability:

*", 0*/ ? 88"8#, 5 .>

where the 8icorrespond to the three patterns of theantecedent.

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The ML14) knowledge engineers found that while an

expert would agree to equation *",+ they becameuneasy and refused to agree with the probability result

*#, 0*/& ? 88"8#, 5 - .> 5 .#

This illustrates these numbers such as .> and .# are

likelihoods of belief+ not probabilities.

3et us have another example.

2uppose this is your last course required for a degree.9ssume your grade-point-average *09, has not beentoo good and you need an O9& in this course to bringup your 09. The following formula may expressyour belief in the likelihood of graduation.

*$, 0*graduating ? O9& in this course, 5 .>

)otice that this likelihood is not P. The reasonit&s not P is that a final audit of your course andgrades must be made by the school. There could be

problem due to a number of reasons that would still

prevent your graduation.

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9ssuming that you agree with *$, *or perhaps your

own value for the likelihood, then by equation *,

*', 0*not graduating ? O9& in this course, 5 .#

From a probabilistic point of view+ *', is correct.

/owever+ it seems intuitively wrong. 4t is %ust notright that if you really work hard and get an O9& in thiscourse+ then there is a #P chance that you won&tgraduate. *', should make you uneasy.

The fundamental problem is that while 0*/ ? 8,

implies a cause of effect relationship between 8 and/+ there may be no cause and effect relationship

between 8 and /&.

These problems with the theory of probability led the theresearchers in ML14) to investigate other ways ofrepresenting uncertainty.

The method that they used with ML14) was based on

certainty &actors.

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Measures o& belie& and disbelie&

4n ML14)+ the certainty factor *1F, was originallydefined as the difference between belief and disbelief.

1F*/+ 8, 5 M=*/+ 8, - M!*/+ 8,

where

1F is the certainty factor in the hypothesis /due to evidence 8

M= is the measure o& increased belie&in / due to 8 M! is the measure o& increased disbelie&in / due to 8

The certainty factor is a way of combining belief and

disbelief into a single number.

1ombining the measures of belief and disbelief into a

single number has some interesting uses.

The certainty factor can be used to rank hypothesis in

order of importance.

For example+ 4f a patient has certain symptomswhich suggest several possible diseases+ then thedisease with the highest 1F would be the one that isfirst investigated by ordering tests.

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The measures of belief and disbelief were defined in termsof probabilities by

5 if 0*/, 5 M=*/+ 8,

maxQ0*/ ? 8,+ 0*/,R - 0*/, 5 ---------------------------------- otherwise - 0*/,

5 if 0*/, 5M!*/+8,

min Q0*/ ? 8,+ 0*/,R - 0*/, 5 ---------------------------------- otherwise - 0*/,

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9ccording to these definitions+ some characteristics

are shown in Table '-.

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS1haracteristics alues

------------------------------------------------------

(anges M=

M!

- 1F

------------------------------------------------------- 1ertain True /ypothesis M= 50*/ ? 8, 5 M! 5

1F 5 -------------------------------------------------------1ertain False /ypothesis M= 5

0*/&?8, 5 M! 5

1F 5 -------------------------------------------------------- 3ack of evidence M= 5 0*/ ? 8, 5 0*/, M! 5

1F 5-------------------------------------------------------

2ome 1haracteristics of M=+ M! and 1F

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The certainty factor+ 1F+ indicates the net belief in

hypothesis based on some evidence.

9 positive 1F means the evidence supports thehypothesis since M= U M!.

9 1F 5 means that the evidence definitely proves

the hypothesis.

9 1F 5 means one of two possibilities.

. First+ a 1F 5 M= - M! 5 could mean that bothM= and M! are .

". The second possibility is that M= 5 M! and bothare nonero. The result is that the belief is

canceled out by the disbelief.

9 negative 1F means that the evidence favors the

negation of the hypothesis since M= V M!. 9notherway of stating this is that there is more reason todisbelief a hypothesis than to belief it.

For example+ a 1F 5 ->P means that the disbelief is>P greater than the belief.

9 1F5>P means that the belief is >P greater thanthe disbelief.

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1ertainty factors allow an expert to express a belief

without committing a value to the disbelief.

The following equation is true.

1F*/+ 8, ; 1F*/&+ 8, 5

The equation means that evidence supporting a

hypothesis reduces support to the negation of thehypothesis by an equal amount so that the sum isalways .

For the example of the student graduating if an O9& is

given in the course

1F*/+8, 5 .> 1F*/&+8, 5 -.>

which means* P certain that 4 will graduate if 4 get

an O9& in this course. *>, 4 am ->P certain that 4 will not graduate if 4

get an O9& in this course.

means no evidence.

2o certainty values greater than favor the hypothesis

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1ertainty factors less than favor the negation of the

hypothesis. 2tatements *, are equivalentusing certainty factors

The above 1F values might be elicited by asking

/ow much do you believe that getting an O9Kwill help you graduateN

if the evidence is to confirm the hypothesis+ or

/ow much do you disbelief that getting O9&will help you graduateN

9n answer of >P to each question will set 1F*/+ 8,5 .>+ and 1F*/&+8, 5 -.>.

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Calculation it% Certainty /actors

9lthough the original definition of 1F was

1F 5 M= - M!

there were difficulties with this definition

because one piece of disconfirming evidence couldcontrol the confirmation of many other pieces ofevidence.

For example+ ten pieces of evidence might produce aM= 5 .CCC and one disconfirming piece with M! 5.>CC could then give

1F 5 .CCC - .>CC 5 ."

The definition of 1F was changed in ML14) in C>>

to be

M= - M!

1F 5 ------------------------ - min*M=+ M!,

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This softens the effects of a single piece of

disconfirming evidence on many confirming pieces ofevidence. Bnder this definition with M=5.CCC+

M!5.>CC

.CCC-.>CC ." 1F 5 --------------------------- 5 ------------- 5 .CC' - min*.CCC+ .>CC, - .>CC

The ML14) method for combining evidence in the

antecedent of a rule are shown in Table '-".

------------------------------------------------------------- 8vidence+ 8 9ntecedent 1ertainty ------------------------------------------------------------- 89)! 8" min Q1F*8+ e,+1F*8"+ e,R

8E( 8" maxQ1F*8+ e,+1F*8"+ e,R)ET 8 -1F*8+ e,

-------------------------------------------------------------- Table '-"

For example+ given a logical expression for

combining evidence such as

8 5 *89)! 8"9)! 8#, or *8$ 9)! )ET 8',

the evidence 8 would be computed as

$#


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8 5 maxQmin*8+ 8"+ 8#,+ min*8$+ -8',Rfor values

85 .C 8"5 .@ 8#5 .#

8$5 -.' 8'5 -.$the result is

8 5 maxQmin*.C+ .@+ .#,+ min*-.'+ -*-.$,R 5 maxQ.#+ -.'R 5 .#

The formula for the 1F of a rule

4f 8 T/8) /is given by

*@, 1F*/+e, 5 1F*8+e, 1F*/+8,

where

1F*8+e, is the certainty factor of the evidence 8making up the antecedent of the rule base onuncertain evidence e.

1F*/+8, is the certainty factor of hypothesisassuming that the evidence is with certainty+ when1F*8+e, 5 .

1F*/+e, is the certainty factor of the hypothesisbased on uncertain evidence e.

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Thus+ if all the evidence in the antecedent is known

with certainty+ the formula for the certainty factor ofthe hypothesis is

1F*/+e, 5 1F*/+8,

since 1F*8+e, 5 .

2ee an example. 1onsider the 1F for the

streptococcus rule discussed before+

4F , The stain of the organism is gram positive+ and", The morphology of the organism is coccus+ and#, The growth confirmation of the organism is chains

T/8) There is suggestive evidence *.>, that the identity of the organism is streptococcus

where the certainty factor of the hypothesis undercertain evidence is

1F*/+ 8, 5 1F */+ 88"8#, 5 .>

and is also called the attenuation &actor.

The attenuation &actor is based on the assumptionthat all the evidence--8+ 8" and 8#--is known withcertainty. That is+

1F*8+ e, 5 1F*8"+ e, 5 1F*8#+ e, 5

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hat happens when all the evidenced are not known

with certaintyN

4n the case of ML14)+ the formula *@, must be usedto determine the resulting 1F value since 1F*/+

88"8#, 5 .> is no longer valid for uncertain

evidence.

For example+ assuming 1F*8+e, 5 .' 1F*8"+e, 5 .< 1F*8#+e, 5 .#

then

1F*8+e, 5 1F*88"8#+e,

5 minQ1F*8+e,+ 1F*8"+e,+ 1F*8#+e,R 5 minQ.'+ .

5 ."

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The following figure summaries the calculations with

certainty factors for two rules based on uncertainevidence and concluding the same hypothesis.

1F of two rules with the same hypothesisbased on uncertain evidence

/ypothesis+ /

1F*/+ e, 5 1F

*8+ e,1F

*/+e, 1F

"*/+ e, 5 1F

"*8+ e,1F

"*/+e,

(ule (ule "

9)! E( )ET*min, *max, *-,

9)! E( )ET*min, *max, *-,

$@


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4n our above example+ if another rule concludes

strepococcus with certainty factor 1F"5 .'+ then thecombined certainty using the first formula of *C, is

1F1EM=4)8*."+ .', 5 ." ; .'* - .", 5 .


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Advantages and disadvantages o&certainty &actors

The 1F formalism has been quite popular with expertsystem developers since its creation because

. 4t is a simple computational model that permitsexperts to estimate their confidence in conclusion

being drawn.

". 4t permits the expression of belief and disbelief ineach hypothesis+ allowing the expression of the effectof multiple sources of evidence.

#. 4t allows knowledge to be captured in a rulerepresentation while allowing the quantification of

uncertainty.

$. The gathering of the 1F values is significantly easierthan the gathering of values for the other methods. )ostatistical base is required - you merely have to ask theexpert for the values.

Many systems+ including ML14)+ have utilied thisformalism and have displayed a high degree ofcompetence in their application areas. =ut is thiscompetence due to these systems& ability to manipulateand reason with uncertainty or is it due to other factorsN

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2ince one purpose of 1F is to rank hypotheses in

terms of likely diagnosis+ it is a contradiction for a

disease to have a higher conditional probability0*/ ? 8, and yet have a lower certainty factor+1F*/+ 8,.

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0 empster,S%a&er *%eory

/ere we discuss another method for handling uncertainty.4t is called !empster-2hafer theory. 4t is evolved duringthe Cs through the efforts of 9rthur!empster and one of his students+ lenn 2hafer.

This theory was designed as a mathematical theory of

evidence.

The development of the theory has been motivated by

the observation that probability theory is not able todistinguish between uncertainty and ignorance owingto incomplete information.

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/rames o& discernment

iven a set of possible elements+ called environment+

5 6+ "+ ...+n7

that are mutually exclusive and exhaustive.

The environment is the set of ob%ects that are of

interest to us.

For example+

5 6airline+ bomber+ fighter7

5 6red+ green+ blue+ orange+ yellow7

Ene way of thinking about is in terms of questions

and answers. 2uppose

5 6airline+ bomber+ fighter7

and the questions is+ Jwhat are the military aircraftNK.

The answer is the subset of

6"+ #7 5 6bomber+ fighter7

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8ach subset of can be interpreted as a possible

answer to a question.

2ince the elements are mutually exclusive and the

environment is exhaustive+ there can be only onecorrect answer subset to a question.

Ef course+ not all possible questions may be

meaningful.

The subsets of the environment are all possible validanswers in this universe of discourse.

9n environment is also called a &rame o&

discernment.

The term discern means that it is possible to

distinguish the one correct answer from all the otherpossible answers to a question.

The power set of the environment *with ")subsets for

a set of sie ), has as its elements all answers to thepossible questions of the frame of discernment.

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Mass /unctions and Ignorance

4n =ayesian theory+ the posterior probability changes asevidence is acquired. 3ikewise in !empster-2hafer theory+the belief in evidence may vary.

4t is customary in !empster-2hafer theory to think

about the degree of belief in evidenceas analogous to

the massof a physical ob%ect.

That is+ the mass of evidence supports a belief.

The reason for the analogy with an ob%ect of mass is

to consider belief as a quantity that can move around+be split up+ and combined.

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9 fundamental difference between !empster-2hafer

theory and probability theory is the treatment ofignorance.

9s discussed in 1hapter $+ probability theory must

distribute an equal amount of probability even inignorance.

For example+ if you have no prior knowledge+ thenyou must assume the probability 0 of each possibility

is

0 5 ----

)

where ) is the number of possibilities.

8.g.+ The formula 0*/, ; 0*/&, 5 must be enforced

The !empster-2hafer theory does not force belief to

be assigned to ignorance or refutation of a hypothesis.

The mass is assigned only to those subsets of theenvironment to which you wish to assign belief.

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9ny belief that is not assigned to a specific subset is

considered nobelie&or nonbelie& and %ust associated

with environment .

=elief that refutes a hypothesis is disbelie&+ which is

not nonbelief.

For example+ we are trying to identify whether anaircraft is hostile. 2uppose there is the evidence of .>indicating a belief that the target aircraft is hostile+where hostile aircraft are only considered to be

bombers and fighters. Thus+ the mass assignment is tothe subset 6bomber+ fighter,+ and

m*6bomber+ fighter7, 5 .>

The rest of the belief is left with the environment+ +as nonbelief.

m*, 5 - .> 5 .#.

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The !empster-2hafer theory has a ma%or difference

with probability theory which would assume that

0*hostile, 5 .>0*non-hostile, 5 - .> 5 .#

.# in !empster-2hafer theory is held as nonbelief in

the environment by m*,. This means neither belief

nor disbeliefin the evidence to a degree of .#.

9 mass has considerably more freedom than probabilitiesas show in the table below.

!empster-2hafer theory 0robability theory-------------------------------------------------------------------

m*, does not have to be 0i5 i

4f A L+ it is not necessary 0*A, 0*L,that m*A, m*L,

)o required relationship 0*A, ; 0*A&, 5 between m*A, and m*A&,

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m"*x, 5 .> if x 56heart#attack+p$lmonary#embolism+aortic#dissection7

otherwise


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Combining evidence

!empster-2hafer theory provides a function forcomputing from two pieces of evidence and theirassociated masses describing the combined influence ofthese pieces of evidence.

This function is known as %empster&s r$le of

combination.

3et mand m"be mass assignments on + the frame of

discernment. The combined mass is computed usingthe formula *special form of empster)s rule o&Combination,

mm"*X, 5 m*A, m"*L,

A L5 X

For instance+ using our hostile aircraft example+ basedon two pieces of evidence+ we obtain

m"*6=7, 5 .C m"*, 5 .

-------------------------------------------------------------m*6=+ F7, 5 .> 6=7 .

m*, 5 .# 6=7 ."> .#

8.g.+ the entry Tis calculated as this


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T*6=7, 5 m*6=+ F7, m"*6=7, 5 *.>,*.C,5. 5 .C =omber

m#*6=+ F7, 5 mm"*6=+ F 7,

5 .> =omber or fighter

m#*, 5 mm"*, 5 .# nonbelief

The m#*6=7, represents the belief that the target is a

bomber and onlya bomber.

m#*6=+ F7, and m#*, imply more information as their

sets include a bomber+ it is plausible that their sumsmay contribute to a belief in the bomber.

2o .> ; .# 5 . may be added to the belief of .C

in the bomber set to yield the maximum belief *5 ,that it could be a bomber. This is called the plausiblebelie&.


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e have two belief values for the bomber+ .C and .

This pair represents a range o& belie&. 4t is called anevidential interval.

The loer boundis called the support*Spt, or (el+

and the upper bound is called plausibility*PIs,.

For instance+ .C is the lower bound in the aboveexample+ and is the upper bound.

The support is the minimum belief based on the

evidence+ while the plausibility is the maximum beliefwe are willing to give.

Thus+ =el 04s . Table below shows some

common evidential interval.

8vidential 4nterval Meaning ------------------------------------------------------------------------

Q+ R 1ompletely true

Q+ R 1ompletely false

Q+ R 1ompletely ignorant

Q=el+ R where V =el V here Tends to support

Q+ 04sR where V 04s V here Tends to refuteQ=el+ 04sR where V =el 04s V here Tends to both support and refute


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The (el*belie& &unction2 or support, is defined to be

the total belief of a set and all its subsets.

=el*A, 5m*L,LA

For example+

=el*6=+ F7, 5 m*6=+ F7, ; m*6=7, ; m*6F7,5 .> ; ; 5 .>

=elief function is different from the mass+ which is the

belief in the evidence assigned to asingleset.

2ince belief functions are defined in terms of masses+

the combination of two belief functions also can beexpressed in terms of masses of a set and all itssubsets.

For example:

=el=el"*6=7, 5 mm"*6=7, ; mm"*,

5 .C ; 5 .C.

=el=el"*6=+ F7,

5 mm"*6=+ F7, ; mm"*6=7, ; mm"*6F7,

5 .> ; .C ; 5 .C>

=el=el"*,

5 mm"*, ; mm"*6=+ F7, ; mm"*6=7,

5 .# ; .> ; .C 5


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=el*, 5 in all cases since the sum of masses must

always equal .

The evidential interval of a set 2+ EI*2,+ may bedefined in terms of the belief.

84*2, 5 Q=el*2,+ - =el*2&,R

For instance+ if 2 5 6=7+ then 2& 5 69+ F7 and

=el*69+ F7,5 mm"*69+ F7, ; mm"*697, ; mm"*6F7,5 ; ; 5

since these are not focal elements and the mass is for nonfocal elements.

Thus+ 84*6=7, 5 Q.C+ - R 5 Q.C+ R

9lso+ since =el*697, 5 + and

=el*6=+ F7, 5 =el=el"*6=+ F7, 5 .C>.

Then84*6=+ F7, 5 Q.C>+ - R 5 Q.C>+ R

84*697, 5 Q+ .#R


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The plausibility is defined as the degree to which the

evidence fails to refute A

04s*A, 5 - =el*A&,

Thus+ 84*A, 5 Q=el*A,+ 04s*A,R

The evidential interval 3total belie&2 plausibility4can

be expressed+

Qevidence for support+ evidence for support ; ignoranceR

The dubiety*bt, or doubtrepresents the degree to

which A is disbelieved or refuted.

The ignorance *Igr, is the degree to which the mass

supports A and A&.

These are defined as follows:

!bt*A, 5 =el*A&, 5 - 04s*A,

4gr*A, 5 04s*A, - =el*A,


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*%e 5ormali#ation o& (elie&

3et us see an example. 2uppose a third evidence now

reports conflicting evidence of an airliner

m#*697, 5 .C'+ m#*, 5 .'

The table shows how the cross products are

calculated.

mm"*6=7, mm"*6=+ F7, mm"*,

.C .> .#----------------------------------------------------------------------------------

m#*697, 5 .C' .@'' .


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mm"m#*A,

5 ."@' ; .$' ; .#' ; .' 5 .>@'

/owever a sum of is required because the combined

evidence mm"m#+ is a valid mass and the sumover all focal elements must be .

This is a problem.

The solution to this problem is a normaliation of the

focal elements by dividing each focal element by

- where is defined for any sets A and L as

5 m*A,m"*L,A L 5

For our problem+

5 .@'' ; .


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The total normalied belief in 6=7 is now

=el*6=7, 5 mm"m#*6=7, 5 .'>#

=el*6=7&, 5 =el*69+ F7,

5 mm"m#*69+ F7, ;

mm"m#*697, ;

mm"m#*6F7,

5 ; .##+ .R

The general form of empster)s Rule o&Combinationis+

m*A, m"*L,A L 5 X

mm"*X, 5 --------------------------

-

)ote that 5 is undefined.

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i&&iculty it% t%e empster,S%a&ert%eory

Ene difficulty with the !empster-2hafer theory occurswith normaliation and may lead to results which arecontrary to our expectation.

The problem is that it ignores the belief that the

ob%ect being considered does not exist.

For example+ the beliefs by two doctors+ 9 and =+ in apatient&s illness are as follows

m9*meningitis, 5 .CC+ m9*brain tumor, 5 .m=*concussion, 5 .CC+ m=*brain tumor, 5 .

=oth doctors think there is a very low chance+ .+ ofa brain tumor but greatly disagree on the ma%or

problem.

The !empster rule of combination gives a combinedbelief of in the brain tumor. The result is very

unexpected

cs511 uncertainty

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