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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)

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

(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

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

- Rolling a fair die
- S 1, 2, 3, 4, 5, 6
- P(E) W(E)/N
- P(3)1/6

- 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

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,

6

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

(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)

(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)

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

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

- 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)

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

(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) ?

Propagation of Belief

P(H1E1E2. En ) P(E1Hi)P(E2Hi). P(EnHi)

P(Hi)

3

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)

(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)

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

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

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

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

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

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

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)

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

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.

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

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

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

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

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

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

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

(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

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)

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

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

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

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

(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