Cs511 Uncertainty (1)

8/20/2019 Cs511 Uncertainty (1)

1/72

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 or ambiguous 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.


2/72

• The uncertainty may also be caused by the represented

knowledge since it might

. (epresent best guesses of the experts that are based 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 of some 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:

"


3/72

• 0robabilistic reasoning.

• 1ertainty factors

• !empster-2hafer Theory

#


4/72

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 the set of all possible outcomes is known. This set of all

possible outcomes of an experiment is known as the sample space of 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 outcome g means that the child is a girland b that 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

$


5/72

• Event: any subset E of the sample space is known as

an event.

• 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"+ ...+

8n in a sample space 2+ are called mutually exclusiveevents if 8i ∩ 8 % 5 ∅+ i ≠ %+ ≤ i+ % ≤ n.

'


6/72

• 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*8 ∪ 8", 5 0*8, ; 0*8",+

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


7/72

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.

>


8/72

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 !a of 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*9 ∩ 9" ∩ ... ∩ 9n, 5

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

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

@


9/72

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 .<

*the probability of a crash and using =rand A,

C


10/72

*'0*A, .@

*>, The probability of a crash+ given that =rand A is notused+ is

0*1 ∩ A&, .

0*1 ? A&, 5 ------------- 5 ------- 5 .'

0*A&, ."

• 0robabilities *', and *


11/72

4n contrast+ the meaning of the conditional probability*


12/72

(ayes) *%eorem

)ote that conditional probability is defined as

0*/ ∩ 8,

0*/ ? 8, 5 ------------------+ for 0*8, ≠ .

0*8,

i.e.+ the conditional probability of / given 8.

• 4n real-life practice+ the probability 0*/ ? 8, cannot

always be found in the literature or obtained fromstatistical analysis. The conditional probabilities

0*8 ? /,

however often are easier to come byD

• 4n medical textbooks+ for example+ a disease is

described in terms of the signs likely to be found in atypical patient suffering from the disease.

• The following theorem provides us with a method for

computing the conditional probability 0*/ ? 8, fromthe probabilities 0*8,+ 0*/, and 0*8 ? /,D

"


13/72

From conditional probability:

0*/ ∩ 8,

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

0*8 ∩ /,Furthermore+ we have+ 0*8 ? /, 5 ---------------

0*/,2o+

0*8 ? /,0*/, 5 0*/ ? 8,0*8, 5 0*/ ∩ 8,Thus0*8 ? /, 0*/,

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

• This is the =ayes& Theorem. 4ts general form can be

written in terms of events+ 8+ and hypotheses*assumptions,+ /+ in the following alternative forms.

0*8 ∩ /i,

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

∑ 0*8 ∩ / %, %

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

∑ 0*8 ? / %, 0*/ %, 0*8, %

#


14/72

$ypot%etical reasoning andbac'ard induction

• =ayes& Theorem is commonly used for decision tree

analysis of business and the social sciences.

• The method of =ayesian decision making is also used

in expert system 0(E2081TE(.

e use oil exploration in 0(E2081TE( as an example.2uppose the prospector believes that there is a better than'-' chance of finding oil+ and assumes the following.

0*E, 5 .< and 0*E&, 5 .$

Bsing the seismic survey technique+ we obtain the

following conditional probabilities+ where ; means a positive outcome and - is a negative outcome

0*; ? E, 5 .@ 0*- ? E, 5 ." *false -,0*; ? E&,5 . *false ;, 0*- ? E&, 5 .C

• Bsing the prior and conditional probabilities+ we can

construct the initial probability tree as shown below.

$


15/72

ProbabilitiesPrior

2ub%ective Epinionof site: 0*/i,

Conditional2eismic test result

0*8?/i,

+oint: 0*8∩/,

50*8?/i,0*/i,

)o oil oil

0*E&, 5 .$ 0*E,5.<

test ; test test ; test0*-?E&, 0*;?E&, 0*-?E, 0*;?E,5.C 5 . 5 ." 5 .@

0*-∩E&, 0*;∩E&, 0*-∩E, 0*;∩E,

5.#< 5.$ 5." 5.$@

4nitial probability tree for oil exploration

'


16/72

• Bsing 9ddition law to calculate the total probability of

a ; and a - test

0*;, 5 0*; ∩ E, ; 0*; ∩ E&, 5 .$@ ; .$ 5 .'"0*-, 5 0*- ∩ E, ; 0*- ∩ E&, 5 ." ; .#< 5 .$@

• 0*;, and 0*-, are unconditional probabilities that can

now be used to calculate the posterior probabilities atthe site+ as shown below.

Probabilitiesunconditional

0*8,

PosteriorEf site:

0*/i?8, 5 0*8?

/i,0*/i,G0*8,

+oint: 0*8∩/,

50*/i?8,0*8,

test ; test0*-, 5 .$@ 0*;,5.'"

)o oil oil )o oil oil0*E&?-, 0*E?-, 0*E&?;, 0*E?;,5#G$ 5 G$ 5 G# 5 "G#

0*-∩E&, 0*-∩E, 0*;∩E&, 0*;∩E,

5.#< 5." 5.$ 5.$@

(evised probability tree for oil exploration


17/72

• The Figure below shows the =aysian decision tree

using the data from the above Figure. The payoffs areat the bottom of the tree.

Thus if oil is found+ the payoff isH+"'+ - H"+ - H'+ 5 H++

while a decision to quit after the seismic test result gives a payoff of -H'+.

The assumed amounts are:Eil lease+ if successful: H+"'+!rilling expense: -H"+2eismic survey: -H'+

Event*est result

; or -

Act.Iuit or drill

EventEil or no oil

payo&&

0*-, 5 .$@ 0*;,5.'"

Iuit !rill Iuit !rill

-H'+ -H'+-H++ H++ -H++ H++

4nitial =ayesian !ecision tree for oil exploration

)o oil oil )o oil oil

0*E&?-, 0*E?-, 0*E&?;, 0*E?;,5#G$ 5 G$ 5 G# 5 "G#

9

1=

! 8

>


18/72

• 4n order for the prospector to make the best decision+

the expected payoff must be calculated at event node9.

• To compute the expected payoff at 9+ we must work

backward from the leaves. This process is calledbac'ard induction.

@


19/72

• The expected payoff from an event node is the sum of

the payoffs times the probabilities leading to the payoffs.

8xpected payoff at node 1H@$


20/72

• The decision tree shows the optimal strategy for the

prospector. 4f the seismic test is positive+ the siteshould be drilled+ otherwise+ the site should be

abandoned.

• The decision tree is an example of hypothetical

reasoning or Jwhat ifK type of situations.

• =y exploring alternate paths of action+ we can prune

paths that do not lead to optimal payoffs.

"


21/72

(ayes) rule and 'noledge,basedsystems

9s we know+ rule-based systems express knowledge in an4F-T/8) format:

4F A is trueT/8) L can be concluded with probability p

• 4f we observe that A is true+ then we can conclude thatL exist with the specified probability. For example

4F the patient has a coldT/8) the patient will sneee *.>',

• =ut what if we reason abductively and observe L *i.e.+the patient sneees, while knowing nothing about A*i.e.+ the patient has a cold,N hat can we concludeabout itN =ayes& Theorem describes how we canderive a probability for A.

"


22/72

• ithin the rule given above+ L *denotes some piece of

evidence *typically referred to as 8, and A denotessome hypothesis */, given

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

0*8,or

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

0*8 ? /,0*/, ; 0*8 ? /&,0*/&,

To make this more concrete+ consider whether (ob has acold *the hypothesis, given that he sneees *the evidence,.

• 8quation *", states that the probability that (ob has a

cold given that he sneees is the ratio of the probability that he both has a cold and sneees+ to the probability that he sneees.

• The probability of his sneeing is the sum of the

conditional probability that he sneees when he has acold and the conditional probability that he sneees

when he doesn&t have a cold. 4n other words+ the probability that he sneees regardless of whether hehas a cold or not. 2uppose that we know in general

""


23/72

0*/, 5 0*(ob has a cold,5 ."

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

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 .'.

"#


24/72

• 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 .>"$<

• 2o knowledge that he sneees increasing his

probability of having a cold by approximately ".'+while knowledge that does not sneee decreases his

probability by a factor of almost #.

"$


25/72

Propagation o& (elie&

)ote that what we have %ust examined is very limitedsince we have only considered when each piece of evidence affects only one hypothesis.

• This must be generalied to deal with JmK hypotheses

/+ /"+ ... /m and JnK pieces of evidence 8+ ...+ 8n+ the

situation normally encountered in real-world problems. hen these factors are included+ 8quation*", becomes

0*8 %∩8 %"∩...∩8 %k ? /i, 0*/i,

*#, 0*/i ? 8 % ∩8 %"∩ ...∩8 %k , 5 -----------------------------------

0*8 % ∩8 %"∩ ...∩ 8 %k ,

0*8 % ? /i,0*8 %" ? /i, ... 0*8 %k ? /i,0*/i,5 -------------------------------------------------------

m

∑0*8 % ? /l,0*8 %" ? /l, ... 0*8 %k ? /l,0*/l,l5

where 6%+ ...+%k , ⊆6+ ...+ n7

• This probability is called the posterior probability of

hypothesis /i from observing evidence 8 %+ 8 %"+ ...+ 8 %k .

"'


26/72

• This equation is derived based on several

assumptions:

. The hypotheses /+ ...+ /m+ m ≥ + are mutuallyexclusive.

". Furthermore+ the hypotheses /+ ...+ /m arecollectively exhaustive.

#. The pieces of evidence 8+ ...+ 8n+ n ≥ + are

conditionally independent given any hypothesis

/i+ ≤ i ≤ m .

Conditional independent: The events 8+ 8"+ ...+8n+ are conditionally independent given an event/ if

0*8 % ∩ ... ∩ 8 %k ? /, 5 0*8 % ? /, ...0*8 %k ? /,

for each subset 6%+ ...+%k , ⊆6+ ...+ n7.

• This last assumption often causes great difficulties for

probabilistic based methods.

For example+ two symptoms+ 9 and =+ might eachindependently indicate that some disease is ' percent

likely. Together+ however+ it might be that thesesymptoms reinforce *or contradict, each other. 1aremust be taken to ensure that such a situation does notexist before using the =ayesian approach.

"


27/72

• To illustrate how belief is propagated through a

system using =ayes& rule+ consider the values shownin the Table below. These values represent

*hypothetically, three mutually exclusive andexhaustive hypotheses

. /+ the patient+ (ob+ has a coldD

". /"+ (ob has an allergyD and

#. /#+ (ob has a sensitivity to light

with their prior probabilities+ 0*/i,&s+ and twoconditionally independent pieces of evidence

. 8+ (ob sneees and

". 8"+ (ob coughs+

which support these hypotheses to differing degrees.

i 5 i 5 " i 5 #*cold, *allergy, *light sensitive,

0*/i, .< .# .0*8 ? /i, .# .@ .#0*8" ? /i, .< .C .

">


28/72

• 4f we observe evidence 8 *e.g.+ the patient sneees,+

we can compute posterior probabilities for thehypotheses using 8quation *#, *where k 5 , to be:

*.#,*.


29/72

0*/" ? 8 ∩ 8",

*.@,*.C,*.#,5 ------------------------------------------------------------

*.#,*.


30/72

Advantages and disadvantages o& (ayesian met%ods

The =ayesian methods have a number of advantages thatindicates their suitability in uncertainty management.

• Most significant is their sound theoretical foundation

in probability theory. Thus+ they are currently the most

mature of all of the uncertainty reasoning methods.

hile =ayesian methods are more developed than theother uncertainty methods+ they are not without faults.

. They require a significant amount of probability datato construct a knowledge base. Furthermore+ human

experts are normally uncertain and uncomfortableabout the probabilities they are providing.

". hat are the relevant prior and conditional probabilities based onN 4f they are statistically based+the sample sies must be sufficient so the probabilitiesobtained are accurate. 4f human experts have provided

the values+ are the values consistent andcomprehensiveN

#. Eften the type of relationship between the hypothesisand evidence is important in determining how theuncertainty will be managed. (educing these

#


31/72

associations to simple numbers removes relevantinformation that might be needed for successfulreasoning about the uncertainties. For example+

=ayesian-based medical diagnostic systems havefailed to gain acceptance because physicians distrustsystems that cannot provide explanations describinghow a conclusion was reached *a feature difficult to

provide in a =ayesian-based system,.

$. The reduction of the associations to numbers also

eliminated using this knowledge within other tasks.For example+ the associations that would enable thesystem to explain its reasoning to a user are lost+ as isthe ability to browse through the hierarchy of evidences to hypotheses.

#


32/72

- Certainty &actors

• 1ertainty factor is another method of dealing with

uncertainty. This method was originally developed for the ML14) system.

• Ene of the difficulties with =ayesian method is that

there are too many probabilities required. Most of

them could be unknown.

• The problem gets very bad when there are many

pieces of evidence.

• =esides the problem of amassing all the conditional

probabilities for the =ayesian method+ another ma%or problem that appeared with medical experts was therelationship of belief and disbelief.

• 9t first sight+ this may appear trivial since obviously

disbelief is simply the opposite of belief. 4n fact+ thetheory of probability states that

0*/, ; 0*/&, 5

and so

#"


33/72

0*/, 5 - 0*/&,

• For the case of a posterior hypothesis that relies on

evidence+ 8

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

• /owever+ when the ML14) knowledge engineers

began interviewing medical experts+ they found that physicians were extremely reluctant to state their knowledge in the form of equation *,.

• For example+ consider a ML14) rule such as the

following.

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

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

• This can be written in terms of posterior probability:

*", 0*/ ? 8∩ 8" ∩ 8#, 5 .>

where the 8i correspond to the three patterns of theantecedent.

##


34/72

• The ML14) knowledge engineers found that while an

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

*#, 0*/& ? 8∩ 8" ∩ 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 reason it&snot 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.

#$


35/72

• 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 not rightthat 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 of representing uncertainty.

• The method that they used with ML14) was based on

certainty &actors.

#'


36/72

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.

#


37/72

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*/,

#>


38/72

• 9ccording to these definitions+ some characteristics

are shown in Table '-.

SSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSSS 1haracteristics 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

#@


39/72

• 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!. 9nother way 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.

#C


40/72

• 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

$


41/72

• 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 -.>.

$


42/72

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 of evidence.

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!,

$"


43/72

• This softens the effects of a single piece of

disconfirming evidence on many confirming pieces of evidence. 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 ------------------------------------------------------------- 8 9)! 8" min Q1F*8+ e,+1F*8"+ e,R

8 E( 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 *8 9)! 8" 9)! 8#, or *8$ 9)! )ET 8',

the evidence 8 would be computed as

$#


44/72

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

8 5 .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 hypothesis based on uncertain evidence e.

$$


45/72

• Thus+ if all the evidence in the antecedent is known

with certainty+ the formula for the certainty factor of the 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 under certain evidence is

1F*/+ 8, 5 1F */+ 8∩8"∩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

$'


46/72

• 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*/+

8∩8"∩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*8∩8"∩8#+e,

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

5 ."

$


47/72

• hat happen when another rule also concludes the

same hypothesis+ but with a different certainty factorN

• The certainty factors of rules concluding the samehypothesis is calculated from the combining functionfor certainty factors defined as

*C, 1F1EM=4)8*1F+1F",

5 1F ; 1F" * - 1F, if both 1F and 1F" U

1F ; 1F"

5 ------------------------- if one of 1F and 1F" V - min*?1F?+?1F"?,

5 1F ; 1F" * ; 1F, if both 1F and 1F" V

where 1F is 1F*/+ e, and 1F" is 1F"*/+ e,.

• The formula for 1F1EM=4)8 used depends on whether

the individual certainty factors are positive or negative.

• The combining function for more than two certainty

factors is applied incrementally. That is+ the 1F1EM=4)8is calculated for two 1F values+ and then the1F1EM=4)8 is combined using formula *C, with the third1F values+ and so forth.

$>


48/72

• 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 hypothesis based 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, *-,

$@


49/72

• 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 .


50/72

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 easier than 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 of competence in their application areas. =ut is thiscompetence due to these systems& ability to manipulateand reason with uncertainty or is it due to other factorsN

'


51/72

2ome studies have shown that changing the certaintyfactors or even turn off the 1F reasoning portion of ML14) does not seems to affect the correct diagnoses

much.

• This revealed that the knowledge described within the

rule contributes much more to the final+ derivedresults than the 1F values.

Ether criticisms of this uncertainty reasoning method

include among others:

. The 1F lack theoretical foundation. =asically+ the 1Fwere partly ad %oc. 4t is an approximation of

probability theory.

". )on-independent evidence can be expressed and

combined only by JchunkingK it together within thesame rule. hen large quantities of non-independentevidence must be expressed+ this proves to beunsatisfactory

#. the 1F values could be the opposite of conditional probabilities.

For example+ if 0*/, 5 .@ 0*/", 5 ."0*/ ? 8, 5 .C 0*/" ? 8, 5 .@

then 1F*/+ 8, 5 .' and 1F*/"+ 8, 5 .>'

'


52/72

• 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,.

'"


53/72

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.

'#


54/72

/rames o& discernment

iven a set of possible elements+ called environment+

Θ 5 6θ+ θ"+ ...+ θn7

that are mutually exclusive and exhaustive.

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

interest to us.

• For example+

Θ 5 6airline+ bomber+ fighter7

Θ 5 6red+ green+ blue+ orange+ yellow7

• Ene way of thinking about Θ is in terms of questions

and answers. 2uppose

Θ 5 6airline+ bomber+ fighter7

and the questions is+ Jwhat are the military aircraftNK.

The answer is the subset of Θ

6θ"+ θ#7 5 6bomber+ fighter7

'$


55/72

• 8ach subset of Θ 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 other possible answers to a question.

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

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

''


56/72

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 evidence as analogous to

the mass of 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.

'


57/72

• 9 fundamental difference between !empster-2hafer

theory and probability theory is the treatment of ignorance.

• 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.

'>


58/72

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

considered no belie& or nonbelie& and %ust associated

with environment Θ.

• =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+ Θ+as nonbelief.

m*Θ, 5 - .> 5 .#.

'@


59/72

• 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*Θ,. This means neither belief

nor disbelief in 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*Θ, does not have to be ∑0i 5 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&,

'C


60/72

e now state things more formally.

• 3et Θ be a frame of discernment *environment,. 9

mass assignment function assigns a number m* x, toeach x ⊆ Θ such that:

*, ≥ m*x, ≥

*", m*∅, 5

*#,∑ m*x, 5 x⊆Θ

• 3et Θ be a frame of discernment *environment,. and

let m be a mass assignment function on Θ. 9 set x ⊆ Θ

is called a focal element in m if m*x, U . The core of m+ denoted by k*m,+ is the set of all focal elements inm.

3et us consider a medical example. 2uppose

Θ 5 6heart-attack+ pericarditis+ pulmonary-embolism+

aortic-dissection7.

• )ote that each mass assignment on Θ assigns mass

numbers to "$

5 < sets. 4f for a specific patient thereis no evidence pointing at a certain diagnosis in

particular+ the mass of is assigned toΘ.

if x 5 Θ


61/72

m*x, 5 otherwise

• 8ach proper subset of Θ gets assigned the number .

The core of m is equal to 6Θ7

• )ow suppose that some evidence has become

available that points to the composite hypothesis

heart#attack or pericarditis with some certainty.

Then the subset 6heart#attack + pericarditis7 will beassigned a mass+ e.g.+ .$. !ue to lack of further information+ the remaining certainty .< is assigned to

Θ.

.< if x 5 Θ

m*x, 5 .$ if x 5 6heart#attack + pericarditis7 otherwise

• )ow suppose we have obtained some evidence

against the hypothesis that our patient is sufferingfrom pericarditis. This information can be consideredas support for hypothesis that the patient is not suffering from pericarditis. This is equivalent to thecomposite hypothesis heart#attack or p$lmonary#

embolism or aortic#dissection. e therefore assign amass+ for example .>+ to the set 6heart#attack + p$lmonary#embolism+ aortic#dissection7

.# if x 5 Θ


62/72

m"*x, 5 .> if x 5 6heart#attack + p$lmonary#embolism+aortic#dissection 7

otherwise


63/72

Combining evidence

!empster-2hafer theory provides a function for computingfrom two pieces of evidence and their associated massesdescribing the combined influence of these pieces of evidence.

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

combination.

• 3et m and m" be mass assignments on Θ+ the frame of

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

m ⊕ m"*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"*Θ, 5 .

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

m*Θ, 5 .# 6=7 ."> Θ .#

8.g.+ the entry T is calculated as this


64/72

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

m#*6=+ F7, 5 m ⊕ m"*6=+ F 7,

5 .> =omber or fighter

m#*Θ, 5 m ⊕ m"*Θ, 5 .# nonbelief

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

bomber and only a bomber.

• m#*6=+ F7, and m#*Θ, 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& .


65/72

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

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

• The loer bound is 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 belief we 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


66/72

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

the total belief of a set and all its subsets.

=el*A, 5∑ m*L,L⊆ A

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 a single set.

• 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 m⊕m"*6=7, ; m⊕ m"*∅,

5 .C ; 5 .C.

=el ⊕ =el"*6=+ F7,

5 m⊕m"*6=+ F7, ; m⊕m"*6=7, ; m⊕ m"*6F7,

5 .> ; .C ; 5 .C>

=el ⊕ =el"*Θ,

5 m⊕ m"*Θ, ; m⊕m"*6=+ F7, ; m⊕m"*6=7,

5 .# ; .> ; .C 5


67/72

• =el*Θ, 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 m⊕m"*69+ F7, ; m⊕m"*697, ; m⊕ m"*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


68/72

• 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 plausibility4 can

be expressed+

Qevidence for support+ evidence for support ; ignoranceR

• The dubiety *bt, or doubt represents 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,


69/72

*%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#*Θ, 5 .'

• The table shows how the cross products are

calculated.

m⊕m"*6=7, m⊕m"*6=+ F7, m⊕m"*Θ,

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

m#*697, 5 .C' ∅ .@'' ∅ .


70/72

∑ m⊕m"⊕m#*A,

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

• /owever a sum of is required because the combined

evidence m⊕m"⊕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

- κ where κ is defined for any sets A and L as

κ 5 ∑ m*A,m"*L,A ∩L 5 ∅

For our problem+

κ 5 .@'' ; .


71/72

• The total normalied belief in 6=7 is now

=el*6=7, 5 m⊕m"⊕m#*6=7, 5 .'>#

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

5 m⊕m"⊕m#*69+ F7, ;

m⊕m"⊕m#*697, ;

m⊕m"⊕m#*6F7,

5 ; .##+ .R

• The general form of empster)s Rule o& Combination is+

∑ m*A, m"*L,A ∩L 5 X

m⊕m"*X, 5 --------------------------

- κ

• )ote that κ 5 is undefined.

>


72/72

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 a patient&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+ .+ of a brain tumor but greatly disagree on the ma%or

problem.

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

unexpected

Documents

Cs511 Uncertainty (1)