Uncertainty in Expert Systems

CPS 4801

Uncertainty• Uncertainty is the lack of exact knowledge

that would enable us to reach a fully reliable solution.oClassical logic assumes perfect

knowledge exists:IF A is trueTHEN B is true

• Describing uncertainty:o If A is true, then B is true with

probability P

Sources of uncertainty• Weak implications: Want to be able to

capture associations and correlations, not just cause and effect.

• Imprecise language: o How often is “sometimes”?o Can we quantify “often,” “sometimes,” “always?”

• Unknown data: In real problems, data is often incomplete or missing.

• Differing experts: Experts often disagree, or have different reasons for agreeing.o Solution: attach weight to each expert

Two approaches• Bayesian reasoning

o Bayesian rule (Bayes’ rule) by Thomas Bayeso Bayesian network (Bayes network)

• Certainty factors

Probability Theory

• The probability of an event is the proportion of cases in which the event occurso Numerically ranges from zero to unity (an

absolute certainty) (i.e. 0 to 1)

P(success) + P(failure) = 1

the number of possible outcomesthe number of successessuccess)P(

the number of possible outcomesfailure)P(

the number of failures

Example• Flip a coin• P(head) = ½ P(tail) = ?• P(head) = ¼ P(tail) = ?

• Throw a dice• P(getting a 6) = ?• P(not getting a 6) = ?

• P(A) = p P(¬A) = 1-p

Example• P(head) = ½ P(head head head) = ?

• Xi = result of i-th coin flip Xi = {head, tail}

• P(X1 = X2 = X3 = X4) = ?

• Until now, events are independent and mutually exclusive.

• P(X,Y) = P(X)P(Y) (P(X,Y) is joint probability.)

Example• P( {X1 X2 X3 X4} contains >= 3 head ) = ?

Conditional Probability

• Suppose events A and B are not mutually exclusive, but occur conditionally on the occurrence of the othero The probability that event A will occur if

event B occurs is called the conditional probability occurcanBtimesofnumberthe

occurcanBandAtimesofnumbertheBAp

probability of A given B

Conditional Probability

• The probability that both A and B occur is called the joint probability of A and B, written p(A ∩ B) Bp

BApBAp

ApABpABp

occurcanBtimesofnumbertheoccurcanBandAtimesofnumbertheBAp

Conditional Probability

• Similarly, the conditional probability that event B will occur if event A occurs can be written as:

BpBApBAp

ApABpABp BpBApBAp

ApABpABp

Conditional Probability

BpBApBAp

ApABpABp

ApABpABp

ApABpBAp

BpBApBAp

ApABpABp

ApABpBAp

BpBApBAp

ApABpABp

ApABpBAp

BpBApBAp

• The Bayesian rule (named after Thomas Bayes, an 18th-century British mathematician):

The Bayesian Rule

BpApABpBAp

Applying Bayes’ rule

• A = disease, B = symptom• P(disease|symptom) = P(symptom|

disease) * P(disease) / P(symptom)

BpApABpBAp

Applying Bayes’ rule• A doctor knows that the disease meningitis

causes the patient to have a stiff neck for 70% of the time.

• The probability that a patient has meningitis is 1/50,000.

• The probability that any patient has a stiff neck is 1%.

• P(s|m) = 0.7• P(m) = 1/50000• P(s) = 0.01

Applying Bayes’ rule• P(s|m) = 0.7• P(m) = 1/50000• P(s) = 0.01• P(m|s) = P(s|m) * P(m) / P(s) • = 0.7 * 1/50000 / 0.01 • = 0.0014• = around 1/714• Conclusion: Less than 1 in 700 patients

with a stiff neck have meningitis.

Example: Coin Flip• P(X1 = H) = ½

1) X1 is H: P(X2 = H | X1 = H) = 0.92) X1 is T: P(X2 = T | X1 = T ) = 0.8

P(X2 = H) = ?

What we learned from the example?

• If event A depends on exactly two mutually exclusive events, B and ¬B, we obtain:

• P(¬X|Y) = 1 – P(X|Y)• P(X|¬Y) = 1 – P(X|Y)?

p(A)=p(AB) p(B)+p(AB) p(B)

p(B) =p(BA) p(A) +p(BA) p(A)

ApABpApABpApABp

BAp

• If event A depends on exactly two mutually exclusive events, B and ¬B, we obtain:

• Similarly, if event B depends on exactly two mutually exclusive events, A and ¬A, we obtain:

Conditional probability

p(A)=p(AB) p(B)+p(AB) p(B)

p(B) =p(BA) p(A) +p(BA) p(A)

ApABpApABpApABp

BAp

p(A)=p(AB) p(B)+p(AB) p(B)

p(B) =p(BA) p(A) +p(BA) p(A)

ApABpApABpApABp

BAp

• Substituting p(B) into the Bayesian rule yields:

The Bayesian Rule

p(A)=p(AB) p(B)+p(AB) p(B)

p(B) =p(BA) p(A) +p(BA) p(A)

ApABpApABpApABp

BAp

BpApABp

BAp

p(A)=p(AB) p(B)+p(AB) p(B)

p(B) =p(BA) p(A) +p(BA) p(A)

ApABpApABpApABp

BAp

• Instead of A and B, consider H (a hypothesis) and E (evidence for that hypothesis).

• Expert systems use the Bayesian rule to rank potentially true hypotheses based on evidences

Bayesian reasoning

HpHEpHpHEpHpHEp

EHp

p(A)=p(AB) p(B)+p(AB) p(B)

p(B) =p(BA) p(A) +p(BA) p(A)

ApABpApABpApABp

BAp

• If event E occurs, then the probability thatevent H will occur is p(H|E)

IF E (evidence) is trueTHEN H (hypothesis) is true with

probability p

Bayesian reasoning

HpHEpHpHEpHpHEp

EHp

Bayesian reasoning Example: Cancer and

Test • P(C) = 0.01 P(¬C) = 0.99• P(+|C) = 0.9 P(-|C) = 0.1• P(+|¬C) = 0.2 P(-|¬C) = 0.8

• P(C|+) = ?

HpHEpHpHEpHpHEp

EHp

Simple Bayes Network from Example

• Expert identifies prior probabilities forhypotheses p(H) and p(¬H)

• Expert identifies conditional probabilities for:o p(E|H): Observing evidence E if hypothesis

H is trueo p(E|¬H): Observing evidence E if

hypothesis H is false

Bayesian reasoningHpHEpHpHEp

HpHEpEHp

• Experts provide p(H), p(¬H), p(E|H), and p(E|¬H)

• Users describe observed evidence Eo Expert system calculates p(H|E) using

Bayesian ruleo p(H|E) is the posterior probability that

hypothesis H occurs upon observing evidence E

• What about multiple hypotheses and evidences?

Bayesian reasoning

Bayesian reasoning with multiple hypotheses

in

ii

n

ii BpBApBAp

11

AB4

B3

B1

B2

p(A)

p(A)=p(AB) p(B)+p(AB) p(B)

p(B) =p(BA) p(A) +p(BA) p(A)

ApABpApABpApABp

BAp

Bayesian reasoning with multiple hypotheses

• Expand the Bayesian rule to work with multiple hypotheses (H1...Hm)

HpHEpHpHEpHpHEp

EHp

Bayesian reasoning with multiple hypotheses and

evidences• Expand the Bayesian rule to work with

multiple hypotheses (H1...Hm) and evidences (E1...En)

• Expand the Bayesian rule to work with multiple hypotheses (H1...Hm) and evidences (E1...En)

Assuming conditional independence among evidences E1...En

Bayesian reasoning with multiple hypotheses and

evidences

m

kkknkk

iiniini

HpHEp...HEpHEp

HpHEpHEpHEpE...EEHp

121

2121

...

Summary

m

kkknkk

iiniini

HpHEp...HEpHEp

HpHEpHEpHEpE...EEHp

121

2121

...

• Expert is given three conditionally independent evidences E1, E2, and E3o Expert creates three mutually exclusive and

exhaustive hypotheses H1, H2, and H3

o Expert provides prior probabilities p(H1), p(H2), p(H3)

o Expert identifies conditional probabilities for observing each evidence Ei for all possible hypotheses Hk

Bayesian reasoning Example

• Expert data:

Bayesian reasoning Example

H ypothesi sProbability

1i 2i 3i0.40

0.9

0.6

0.3

0.35

0.0

0.7

0.8

0.25

0.7

0.9

0.5iHp

iHEp 1

iHEp 2

iHEp 3

• user observes E3 H ypothesi sProbability

1i 2i 3i0.40

0.9

0.6

0.3

0.35

0.0

0.7

0.8

0.25

0.7

0.9

0.5iHp

iHEp 1

iHEp 2

iHEp 3

Bayesian reasoning Example

32,1,=,3

13

33 i

HpHEp

HpHEpEHp

kkk

iii

0.3425.0.90+35.07.0+0.400.6

0.400.631

EHp

0.3425.0.90+35.07.0+0.400.6

35.07.032

EHp

0.3225.0.90+35.07.0+0.400.6

25.09.033

EHp

32,1,=,3

13

33 i

HpHEp

HpHEpEHp

kkk

iii

0.3425.0.90+35.07.0+0.400.6

0.400.631

EHp

0.3425.0.90+35.07.0+0.400.6

35.07.032

EHp

0.3225.0.90+35.07.0+0.400.6

25.09.033

EHp

32,1,=,3

13

33 i

HpHEp

HpHEpEHp

kkk

iii

0.3425.0.90+35.07.0+0.400.6

0.400.631

EHp

0.3425.0.90+35.07.0+0.400.6

35.07.032

EHp

0.3225.0.90+35.07.0+0.400.6

25.09.033

EHp

32,1,=,3

13

33 i

HpHEp

HpHEpEHp

kkk

iii

0.3425.0.90+35.07.0+0.400.6

0.400.631

EHp

0.3425.0.90+35.07.0+0.400.6

35.07.032

EHp

0.3225.0.90+35.07.0+0.400.6

25.09.033

EHp

expert system computes

posterior probabilities

user observes E3

H ypothesi sProbability

1i 2i 3i0.40

0.9

0.6

0.3

0.35

0.0

0.7

0.8

0.25

0.7

0.9

0.5iHp

iHEp 1

iHEp 2

iHEp 3

• user observes E3 E1 H ypothesi sProbability

1i 2i 3i0.40

0.9

0.6

0.3

0.35

0.0

0.7

0.8

0.25

0.7

0.9

0.5iHp

iHEp 1

iHEp 2

iHEp 3

32,1,=,3

131

3131 i

HpHEpHEp

HpHEpHEpEEHp

kkkk

iiii

0.1925.00.5+35.07.00.8+0.400.60.30.400.60.3

311

EEHp

0.5225.00.5+35.07.00.8+0.400.60.335.07.00.8

312

EEHp

0.2925.00.5+35.07.00.8+0.400.60.325.09.00.5

313

EEHp

Bayesian reasoning Example

user observes E1

32,1,=,3

131

3131 i

HpHEpHEp

HpHEpHEpEEHp

kkkk

iiii

0.1925.00.5+35.07.00.8+0.400.60.30.400.60.3

311

EEHp

0.5225.00.5+35.07.00.8+0.400.60.335.07.00.8

312

EEHp

0.2925.00.5+35.07.00.8+0.400.60.325.09.00.5

313

EEHp

expert system computes

posterior probabilities

H ypothesi sProbability

1i 2i 3i0.40

0.9

0.6

0.3

0.35

0.0

0.7

0.8

0.25

0.7

0.9

0.5iHp

iHEp 1

iHEp 2

iHEp 3

m

kkknkk

iiniini

HpHEp...HEpHEp

HpHEpHEpHEpE...EEHp

121

2121

...

• user observes E3 E1 E2 H ypothesi sProbability

1i 2i 3i0.40

0.9

0.6

0.3

0.35

0.0

0.7

0.8

0.25

0.7

0.9

0.5iHp

iHEp 1

iHEp 2

iHEp 3

32,1,=,3

1321

321321 iHpHEpHEpHEp

HpHEpHEpHEpEEEHp

kkkkk

iiiii

0.4525.09.00.50.7

0.7

0.7

0.5

+.3507.00.00.8+0.400.60.90.30.400.60.90.3

3211

EEEHp

025.09.0+.3507.00.00.8+0.400.60.90.335.07.00.00.8

3212

EEEHp

0.5525.09.00.5+.3507.00.00.8+0.400.60.90.325.09.00.70.5

3213

EEEHp

32,1,=,3

1321

321321 iHpHEpHEpHEp

HpHEpHEpHEpEEEHp

kkkkk

iiiii

0.4525.09.00.50.7

0.7

0.7

0.5

+.3507.00.00.8+0.400.60.90.30.400.60.90.3

3211

EEEHp

025.09.0+.3507.00.00.8+0.400.60.90.335.07.00.00.8

3212

EEEHp

0.5525.09.00.5+.3507.00.00.8+0.400.60.90.325.09.00.70.5

3213

EEEHp

Bayesian reasoning Example

expert system computesposterior probabilitiesuser observes E2

H ypothesi sProbability

1i 2i 3i0.40

0.9

0.6

0.3

0.35

0.0

0.7

0.8

0.25

0.7

0.9

0.5iHp

iHEp 1

iHEp 2

iHEp 3

m

kkknkk

iiniini

HpHEp...HEpHEp

HpHEpHEpHEpE...EEHp

121

2121

...

Bayesian reasoning Example

• Initial expert-based ranking:o p(H1) = 0.40; p(H2) = 0.35; p(H3) = 0.25

• Expert system ranking after observing E1, E2, E3:o p(H1) = 0.45; p(H2) = 0.0; p(H3) = 0.55

H ypothesi sProbability

1i 2i 3i0.40

0.9

0.6

0.3

0.35

0.0

0.7

0.8

0.25

0.7

0.9

0.5iHp

iHEp 1

iHEp 2

iHEp 3

Problems with the Bayesianapproach

• Humans are not very good at estimating probability!o In particular, we tend to make different

assumptions when calculating prior and conditional probabilities

• Reliable and complete statistical information often not available.

• Bayesian approach requires evidences to be conditionally independent – often not the case.

• One solution: certainty factors