Chapter 6' Uncertainty Management

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Chapter 6. Uncertainty Management
Recap
- We do not know if some event is T or F with
complete certainty, rather we have some degree
of belief in event Uncertain facts It probably
will rain today. Uncertain rules If it is
cloudy, then it will probably rain.
Overview
- Uncertainty problem - Bayesian Approach (based
on probability theory) - Certainty Factors (CF)
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1. Uncertainty Problem
- Most intelligent tasks have some degree of
uncertainty. - KBS exhibit intelligent behaviour
therefore natural expectation that they can
reason with uncertainty. (e.g., Mycin,
CF(condition)gt0.2)
(1) Problems Encountered - Data - Missing,
unavailable - Unreliable, ambiguous, conflict -
Representation imprecise, inconsistent - Best
guess, based on default (exceptions) - Heuristic
rules
Rule's conclusion not guaranteed to be
correct Reasoning from observed symptoms to their
possible causes - abductive
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(2) KBS Considerations - How to represent
uncertain data - How to combine two or more
pieces of uncertain data - How to draw inference
using uncertain data
(3) Different Approaches
- Four numerically oriented methods Bayes
Rules Certainty Factors Dempster Shafer Fuzzy
Sets - Quantitative approaches Non-monotonic
reasoning - Symbolic approaches Cohens Theory
of Endorsements Foxs semantic systems
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2. Bayesian Approaches
- Bayes Rule oldest best defined technique
for managing uncertainty
(1) Probability Theory - basic concepts
- Probability P(E) exist a number P(E) called
probability, which is the likelihood of event E
occurring from a random experiment e.g.
P(H) P(T) 50 (the game of throwing a coin,
Hhead, Ttail) - Sample space S set of all
possible outcomes of an experiment is called
sample space S S S1, S2, Si, Sn), each
possible outcome Si is an event E and is part
of the sample space. P(Si) W(Si)/N P(Si)
probability of event Si (likelihood of Si
happening) W(Si) number of event Si happened
during experiments N total number of the
experiments
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• Rolling a fair die
• S 1, 2, 3, 4, 5, 6
• P(E) W(E)/N
• P(3)1/6

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- Probability is a number which predicts the
expected likelihood of a given event from an
experiment and is constrained by the following
relationships 0 lt P(E) lt 1
P(E) P(E) 1 E event E not happening
(2) Compound Probabilities - combinations of
different events, e.g., probability of two
different events occurring or either of them
occurring
- Intersection Problems concerned with multiple
events, we must first determine the
intersection of the sample spaces ----gt joint
probability
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Joint probability of two independent events A
and B occurring P(A and B)
P(AB) n(AÇB) P(A)P(B)
n(S)
Example the probability of rolling an odd-number
die and also one divisible by 3 Sample space S
1,2,3,4,5,6 n(S) 6 Event A odd-number
A 1,3,5 Event B divisible by 3 B 3,
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P(A B) 1/6
Ç
n(A Ç B) 1
P(A) W(A)/N 3/6 1/2 P(B) W(B)/N 2/6
1/3
P(A B) P(A)P(B) 1/21/3 1/6
Ç
- Union Problems concerned with determining
the probability of either one or several
events occurring, P(A or B)
P(AÈB)P(A) P(B) - P(AÇB)
Dice example P(A B) 1/2 1/3 - 1/6 2/3
È
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(3) Conditional Probabilities - events that are
not mutually exclusive will influence one
another. - Given that an event B has already
occurred, the probability of an event A
occurring is called the conditional probability
P(AB)P(AÇB)
P(B)
Dice example event B number divisible by 3 has
occurred, the probability of event A
rolling number 3 is
P(AB) n(AÇ B) n(B) n(AÇ B) 1
n(S)
2
n(S)
n(B)
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(4) Bayesian Theory
- Conditional probability can obtain the
probability of an event A given that B has
occurred, BUT often concerned with the reverse
(forward in time) - APosteriori probability the
probability of an earlier event given some
later ones has occurred (backward in time) -
Bayes Theorem probability of the truth of some
hypothesis H given some evidence E
P(HE) P(H)P(EH)
P(E)
P(HE) probability that H is true given evidence
E P(H) probability that H is true P(EH)
probability of observing evidence E when H is
true P(E) probability of E where P(E)
P(EH)P(H) P(EH)P(H)
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P(EH) probability that E is true when H is
false P(H) probability that H is false
P(HE) P(H)P(EH)
P(EH)P(H) P(EH)P(H)
- Bayess theorem relies on knowing prior
probability of an event then used to interpret
present situation - Used for KBS design IF E
THEN H Example consider whether Bob has a cold
(H) given that he sneezes (E).
P(HE) P(H)P(EH)
P(E) P(EH)P(H) P(EH)P(H)
P(E)
Probability that Bob has a cold he sneezes
probability he has a cold sneezes/
probability that he sneezes Probability of
his sneezing he sneezes when he has a cold he
sneezes when he doesnt have a cold
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P(H) P(Bob has a cold) 0.2 P(EH)
P(Bob was observed sneezing Bob has a cold)
0.75 P(EH) P(Bob was observed sneezing Bob
does not have a cold) 0.2 then P(E)
P(EH)P(H) P(EH)P(H) 0.750.2
0.2(1-0.2) 0.31 and P(HE) P(Bob
has a cold Bob was observed sneezing)
P(EH)P(H)/P(E) 0.750.2/0.31
0.5 P(HE) P(Bob has a cold Bob was not
sneezing) P(EH)P(H)/P(E)
(1-0.75)0.2/(1-0.31) 0.07
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• Patients with chest pains are given an ECG
• Results are (ECG) suggesting (HD) or (-ECG)
  suggesting (-HD)
• What is the probability of a patient with a ECG
  having HD? P(HD)ECG) P(HE)
• Assume the following probabilities
• P(HD) 0.1 1 person in 10 has heart disease
  P(H)
• P(-HD) 0.9 (1-0.1) P(H)
• P(ECGHD)0.9 9/10 people with HD have a ve
  ECG. P(EH)
• P(-ECG-HD)0.95 9.5/10 who do not have HD have
  -ve P(EH)
• P(ECG-HD)0.05 5/100 who dont have HD have ve
  P(EH)
• 1-P(-ECG-HD)

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P(HE) P(H)P(EH)
P(EH)P(H) P(EH)P(H)
P(HDECG) 0.90.1
(0.90.1)(0.050.9) 0.67 2/3 of
people with a ve ECG have heart disease
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(5) Propagation of Belief
- Deal with m hypotheses and n evidence -
Posterior probability of hypothesis Hi from
observing evidence E1,,En. (evidence
conditionally independent given hypothesis)
P(H1E1E2. En ) P(E1Hi)P(E2Hi). P(EnHi)
P(Hi)
m
S P(E1Hk) P(E2Hk). P(EnEk) P(Hk)
k1
Example three hypothesis and two conditionally
independent pieces of evidence E1
sneezes E2 coughs
i 1 i 2 i 3 cold allergy light sensitive
P(Hi) 0.6 0.3 0.1 P(E1Hi) 0.3
0.8 0.3 P(E2Hi) 0.6 0.9 0.0
P(H1E1E2) ? P(H2E1E2) ? P(H3E1E2) ?
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Propagation of Belief
P(H1E1E2. En ) P(E1Hi)P(E2Hi). P(EnHi)
P(Hi)

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S P(E1Hk) P(E2Hk). P(EnEk) P(Hk)
k1
0.30.60.6
P(H1E1E2)
(0.30.60.6)(0.80.90.3)(0.30.00.1)
0.80.90.3
P(H2E1E2)
(0.30.60.6)(0.80.90.3)(0.30.00.1)
0.30.00.1
P(H3E1E2)
(0.30.60.6)(0.80.90.3)(0.30.00.1)
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(6) Advantages Disadvantages
- Advantages - Sound theoretical foundation in
probability theory - Well-defined semantics for
decision making - Disadvantages - Require a
significant amount of probability data all
the data on the relationships of the evidence
with various hypotheses must be known all
the relationships between evidence and
hypothesis P(EH) must be independent - What
are the relevant prior and conditional
probabilities based on? - Lack of
explanations (reduce relationships between H
E to simple numbers)
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3. Certainty Theory

• Alternative to Bayes theory
• Grew out of work done on MYCIN
• Relies on judgmental measures of belief as
  opposed to strict probability estimates
• Do you have a severe headache
• subjective
• probability scale 0 -1 answer 0.7
• what does this represent - how do we reason from
  this?
• no statistical basis - cant use Bayes

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Certainty Theory

• MYCIN was developed to diagnose blood disorders
  and their treatment.
• Had to develop a means to deal with
  inexact/incomplete information - typical of
  medical domains
• Often patients require urgent treatment
• Information often incomplete/missing or
  inaccessible (eg allergic to drug/certain foods!)
• Test results which are important for diagnosis
  may take time - culture growth of bacteria
• Doctor may have to act fast based on current
  information

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Certainty Theory

• Decide what organism/organisms may be present
  causing infection
• How best to treat them
• Bayes inappropriate - no statistical information
• Devised a CF range -1 to 1
• -ve values indicate degree of disbelief
• ve values indicate degree of belief

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Certainty Theory

• Eg of MYCIN rule
• IF the stain of the organism is gram pos
• AND the morphology of the organism is coccus
• AND the growth of the organism is chains
• THEN there is evidence that the organism is
  streptococcus CF 0.7
• Given the evidence a doctor only partially
  believe the conclusion
• General Form
• IF E1 And E2 .THEN H CF Cfi
• where E evidence H is the conclusion

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Certainty Theory

• Uncertain Inferencing
• if belief in the evidence is not certain then
  belief in a related inference is decreased
• Combining evidence from multiple sources
• when receive supporting information from a number
  of sources then belief increased
• 1) If A And B Then Z CF 0.8
• 2) If C And D Then Z CF 0.7
• Same Hypothesis, different CF from different
  evidence
• Firing both rules leads to 2 beliefs

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Certainty Theory

• How do we combine these 2 beliefs in Z?
• MYCIN team decided to increase belief towards 1
  as more and more confirming evidence was
  gathered.
• CF would be partially incremented
• Net Belief
• during evidence gathering for a hypothesis some
  evidence adds to belief while other pieces of
  evidence detracts from it
• doctor weights up all evidence to get net belief
• supporting evidence - evidence of belief (MB)
• rejecting evidence - measure of disbelief (MD)
• CF MB-MD

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Certainty Theory

• P(H) prior probability of hypothesis H - experts
  belief
• P(H) experts disbelief
• If expert observes evidence such that the
  probability of the hypothesis is greater than
  prior probability then the experts belief
  increases according to
• P(HE)-P(H)
• If expert observes evidence such that the
  probability of the hypothesis is less than
  prior probability then the experts belief
  decreases according to
• P(H) - P(H\E)

- measure of change in belief
1-P(H)
- measure of change in belief
P(H)
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Certainty Theory

• These measures of change in belief gave rise to
  MB MD
• Measure of Belief (MB)- a number which reflects
  the measure of increased belief in a hypothesis H
  based on evidence E
• MB(H,E)

If P(H)1
1
maxP(HE),P(H)-P(H)
1-P(H)
P(H) prior probability of H being true P(HE)
probability of H being true given evidence E
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Certainty Theory

• Measure of Disbelief (MD)- a number which
  reflects the measure of increased disbelief in a
  hypothesis H based on evidence E
• MD(H,E)

If P(H)0
1
minP(HE),P(H)-P(H)
-P(H)
P(H) prior probability of H being true P(HE)
probability of H being true given evidence E As
several pieces of information may be observed
(some adding to and some detracting from the
belief) a third measure is introduced to combine
the MB MD.
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Certainty Theory

• Certainty Factor
• CF MB-MD
• -1ltCFlt1
• -1 means definitely false
• 1 definitely true
• zero unknown

-1
0
1 T
F
range of disbelief range of belief
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Certainty Theory

• Statement - It will rain today CF 0.6
• CF 0.6 represents the degree of belief in
  statement
• CF are not probabilities but informal measures of
  confidence
• Rule - If it is cloudy then it will rain CF 0.8
• represents the relationship between the evidence
  in the rules premise and the hypothesis in its
  conclusion
• Certainty Propagation
• Concerned with establishing the level of belief
  in a rules conclusion when the available evidence
  in the rules premise in uncertain
• Multiply the CF of the premise with that of the
  rule

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Certainty Propagation

• Certainty Propagation for Single Premise Rules
• If it is cloudy CF 0.5
• then it will rain CF 0.8
• CF(H,E) 0.50.8 0.4
• (If premise CF -0.5 then CF rule -0.4)
• Certainty Propagation for Multiple Premise Rules
  in MYCIN
• Two methods for propagation
• conjunctive rules
• disjunctive rules

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Certainty Propagation for Multiple Premise Rules
in MYCIN

• Conjunctive rules
• If the sky is dark CF 1.0
• and the wind is strong CF 0.7
• then it will rain CF 0.8
• CF(conclusion) minCF(evidence) CF(rule)
• CF(It will rain) Min1.0,0.70.8
• CF(It will rain) 0.56

evidence
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Certainty Propagation for Multiple Premise Rules
in MYCIN

• Disjunctive rules
• If the sky is dark CF 1.0
• or the wind is strong CF 0.7
• then it will rain CF 0.8
• CF(conclusion) maxCF(evidence) CF(rule)
• CF(It will rain) Max1.0,0.70.8
• CF(It will rain) 0.8

evidence
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Certainty Propagation for Similarly Coded Rules
in MYCIN

• Sometimes there is a need to consider multiple
  rules for a particular hypothesis
• If the weather man says it is going to rain
• then it will rain CF 0.8
• If the farmer says it is going to rain
• then it will rain CF 0.9
• If we receive supporting evidence for a
  conclusion from 2 different sources we are more
  confident about it
• MYCIN team developed a technique to combine CF
  values established by rules concluding the same
  hypothesis

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Certainty Propagation for Similarly Coded Rules
in MYCIN

• CFrevised(CFold, CFnew)
• CFold CFnew(1-CFold) if both CFold and CFnew
  gt0
• CFold, CFnew (1CFold ) if both CFold and
  CFnew lt0
• if one of CFold and CFnew lt0

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(4) Example of CFs Propagation
CFold0.0 CFnew0.675
Guilty CF 0.0
Guilty CFrevised0.675
CFcon1CFnew0.675
CFrevisedCFold CFnew(1-CFold) 0.0
0.675(1-0.0) 0.675
fingerprints on weapon CFevid10.90 CFrule10.75
RULE 1. IF the defendants fingerprints are on
the weapon THEN the defendant is guilty
CFcon1CFevid1CFrule1 (single premise rule)
0.90.75 0.675
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CFold0.675 CFnew0.30
Guilty CFrevised0.772
Guilty CFrevised0.675
CFcon2CFnew0.30
CFrevisedCFold CFnew(1-CFold) 0.675
0.30(1-0.675) 0.7725
Motive exists CFevid20.50 CFrule20.60
RULE 2. IF the defendant has a motive THEN
the defendant is guilty of the crime
CFcon2CFevid2CFrule2 0.500.60 0.30
(single premise rule)
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CFold0.772 CFnew-0.76
CFcon3CFnew-0.76
Guilty CFrevised0.772
Guilty CFrevised0.052
CFreviced
Alibi found CFevid30.95 CFrule3 -0.80
(0.772-0.76)/(1-0.76) 0.052
RULE 3. IF the defendant has an alibi THEN he
is not guilty
CFcon3CFevid3CFrule3 0.95(-0.80)
-0.76
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Example of CFs Propagation

• 1. If the weather man says it is going to rain
• then it will rain CF 0.8
• 2. If the farmer says it is going to rain
• then it will rain CF 0.8
• Case 1 Both are certain it will rain
• Cfold 1.00.8 0.8
• Cfnew 1.00.8 0.8
• CFold CFnew(1-CFold)
• 0.80.8(1-0.8)
• 0.96

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Example of CFs Propagation

• Case 2 Weatherman certain of rain farmer
  certain no rain
• Cfold 1.00.8 0.8
• Cfnew -1.00.8 -0.8
• 0.8-0.8/1-min(0.8,-0.8)
• 0
• Cancel each other out ie unknown

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Example of CFs Propagation

• Case 3 Weatherman farmer believe in no rain to
  different degrees
• Cfold -0.80.8 -0.64
• Cfnew -0.60.8 -0.48
• CFold CFnew (1CFold )
• -0.64-0.48(1-0.64)
• -0.81

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(5) Advantages Disadvantages
- Advantages - Simple computational model -
Allows experts estimate confidence in
conclusions - Allows expression of belief,
disbelief effect of multiple sources of
evidence - Knowledge captured in rule
representation quantification of
uncertainty - Gathering CF easy - ask expert -
Disadvantages - Nonindependent evidence must be
chunked within rule - Modification of KB complex