1. Dealing with Uncertainty

2. Reasoning Under Uncertainty
Monotonic Reasoning
A reasoning process that moves in one direction only.
The number of facts in the knowledge base is always increasing.
The conclusions derived are valid deductions and they remain so.
Reasoning process applied to practical everyday problems must recognize uncertainty
Available information is frequently incomplete
Conditions change over time
There is frequently a need to make an efficient but possibly incorrect guess when reasoning reaches a dead end.

3. Reasoning Under Uncertainty
Non-monotonic Reasoning
Non-monotonic reasoning (NMR) is based on augmenting absolute truth with beliefs.
These tentative beliefs are generally based on default assumptions that are made in light of lack of evidence.
A NMR system tracks a set of tentative beliefs and revise those beliefs when knowledge is observed or derived.

4. Reasoning Under Uncertainty
Uncertainty may cause bad treatment in medicine, loss of money in business.
Classic examples of successful expert systems which deal with uncertainty are MYCIN for medical diagnosis and PROSPECTOR for mineral exploration.
In case of medicine, delaying treatment for more tests (for more exact knowledge) may add considerable costs; the patient may die.

5. Reasoning Under Uncertainty
Many different types of errors can contribute to uncertainty.
data might be missing or unavailable
data might be ambiguous or unreliable due to measurement errors
the representation of data may be imprecise or inconsistent
data may just be user's best guess (random)
data may be based on defaults, and defaults may have exceptions

6. Reasoning Under Uncertainty
Given these sources of errors, most knowledge base systems incorporate some form of uncertainty management.
There are three issues to be considered:
How to represent uncertain data.
How to combine two or more pieces of uncertain data.
How to draw inference using uncertain data

7. Reasoning Under Uncertainty
Errors and Induction
Deduction is going from general to specific
All men are mortal
Socrates is a man
therefore Socrates is mortal
Induction tries to generalize from the specific.
My disk has never crashed.
Inductive
therefore my disk will never crash.

8. Reasoning Under Uncertainty
Inductive arguments can never be proven correct (except mathematical induction). Inductive arguments can provide some degree of confidence that the conclusion is correct.
Deductive errors or fallacies may also occur
If p implies q
q is true
therefore p
Example
If the valve is in good condition then the output is normal
The output is normal
Therefore, the valve is in good condition
Uncertainty is major problem in knowledge elicitation, especially when the expert's knowledge must be quantized in rules.

9. Approaches in Dealing with Uncertainty
Numerically oriented methods:
Bayes’ Rules
Certainty Factors
Dempster Shafer
Fuzzy Sets
Quantitative approaches
Non-monotonic reasoning
Symbolic approaches
Cohen’s Theory of Endorsements
Fox’s semantic systems

10. Classical Probability
This is also called a priori probability. It is assumed that all possible events are known and that each event is equally likely to happen (rolling a die).
Prior or unconditional probability is the one before the evidence is received.
Posterior or conditional probability is the one after the evidence is received.

11. Theory of Probability
Formal theory of probability can be made using 3 axioms:
Axiom 1
0 =< P(E) =< 1
Axiom 2
where Ei, i=< 1=< n are mutually
exclusive
P(E) + P(E') = 1
Axiom 3

12. P(E) =[ lim (N-> infinity)] f(E)/N
Theory of Probability
Experimental or Subjective Probabilities
In contrast to the prior approach, experimental probability defines the probability of an event P(E) as the limit of a frequency distribution.
P(E) =[ lim (N-> infinity)] f(E)/N
This type of probability is called a posterior probability.
A subjective probability is a belief or opinion expressed as a probability rather than a probability based on axioms or empirical measurements. This is applied on the decisions for non-repeatable events.

13. Theory of Probability Compound Probabilities
What is the probability of rolling a die with an outcome of even number divisible by 3.
Event A=
{2, 4, 6}
Event B =
{3, 6}

14. Theory of Probability
The two events are called stochastically independent events if and only if the above formula is true.
Stochastic is a Greek word meaning "guess". It is commonly used as a synonym for probability, random or chance.
The probability of rolling a die with an outcome of even number or divisible by 3.
= 3/6 + 2/6 – 1/6 = 4/6

15. Theory of Probability Conditional Probabilities
The probability of an event A, given that event B occurred, is called a conditional probability and indicated by P(A|B).

16. Baye's Theorem
Baye's Theorem in terms of events E, and hypothesis,

17. Baye's Theorem
The conditional probability, P(A|B), states the probability of event A given that event B occurred. The inverse problem is to find the inverse probability which states the probability of an earlier event given that a later one occurred.
Example: Probability of chosing brand X given it has crashed.
This is inverse or posterior probability.

18. Example X X’ Total of Rows Crash C No Crash C’ 0.6 0.1 0.2 0.1 0.7 0.3
Table below shows hypothetical disk crashes using a brand X drive within one year
X X’
Total of Rows
Crash C
No Crash C’
0.7
0.3
Total of Columns
1.0
P(C|X) = ?
P(C|X) = P(C  X) / P(X) = 0.6 / 0.8 = 0.75
P(C|X’) = P(C  X’) / P(X’) = 0.1 / 0.2 = 0.50
P(X|C) = ?
P(X|C) = P(C  X) / P(C) = 0.6 / 0.7 = 6/7
P(X|C) = P(C|X) P(X) / P(C) = 0.75 * 0.8 / 0.7
= 0.6 / 0.7

19. Example
Suppose, statistics show that Brand X drive crashes with a probability of 75% within one year and non-Brand X drive crash within one year is 50%. The inverse question is, what is the probability of a crashed drive being brand X or non-brand X.

20. Hypothetical Reasoning and Backward Induction
Bayesian decision making is used in PROSPECTOR to decide favorable sites for mineral exploration.
Generally conditional probability is forward in time, while a posterior probability is backward in time.
Example of Bayesian decision making under uncertainty.

21. Oil exploration
If there is no evidence for or against we may guess that
P(O) = P(O') = 0.5
We may believe that the chances are better for finding oil.
P(O) = 0.6, P(O') = 0.4
Assume the probabilities for the outcomes of seismic test for oil exploration as:
P(+|O) = 0.8, P(-|O) = 0.2 (false -)
P(+|O')= 0.1 (false +), P(-|O') = 0.9

22. Using above conditional (prior) probabilities we can construct the initial probability tree .

23. .

24. Advantages and Disadvantages of Bayesian Methods
Bayesian methods have support of probability theory and have well defined semantics for decision making.
Disadvantages are
They require significant amount of probability data to construct a knowledge base.
If the probabilities are statistical, sample size must be sufficient. If they are provided by an expert then their comprehensiveness and consistency must be queried.
Reducing associations between the hypotheses and the evidences to simple numerical values removes relevant information necessary for reasoning (explanation of how a conclusion is reached).

25. Reasoning with Certainty Factors
During the development of MYCIN, researchers developed certainty factors formalism for the following reasons:
The medical data lacks large quantities of data and/or the numerous approximations required by Bayes' theorem.
There is a need to represent medical knowledge and heuristics explicitly, which can not be done when using probabilities.
Physicians reason by capturing evidence that supports or denies a particular hypothesis.

26. Certainty Factor (CF) Formalism
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

27. Certainty Factor (CF) Formalism
A measure of belief, MB(h, e) indicates the degree to which our belief in hypothesis, h, is increased based on the presence of evidence, e
A measure of disbelief, MD(h, e), indicates the degree to which our disbelief in hypothesis, h, is increased based on the presence of evidence, e.
When
p(h | e) = 0
MB(h, e) = 0
MD(h, e) = 1
p(h | e) = 1
MB(h, e) = 1
MD(h, e) = 0

28. Certainty Factor (CF) Formalism
CF interpretation

29. Certainty Factor (CF) Formalism
CF(h | e) = MB(h, e) - MD(h, e)
 -1 < CF < 1 
When there is total belief
CF = 1, and
When there is a total disbelief in hypothesis
CF = -1
When there is no evidence to make judgment
CF = 0 F -1 1 T
range of disbelief range of belief

30. Certainty Factor (CF) Formalism
Composite CF can be calculated as follows:
CFcomp(h, e) = MBcomp(h, e) - MDcomp(h, e) 
For P1 and P2 premises of the rule,
CF(P1 and P2)= MIN((CF(P1), CF(P2))
CF(P1 or P2) = MAX ((CF(P1), CF(P2))

31. Certainty Factor (CF) Formalism
For example consider a rule in a knowledge base:
(P1 and P2) or P3 ® R1(.7) and R2(.3)
If CFs for P1, P2, and P3 are 0.6, 0.4, and 0.2, respectively then R1 and R2 may be anticipated with CFs 0.28 and 0.12 respectively.
CF(P1(0.6) and P2(0.4)) = MIN(.6, .4) = 0.4
CF((0.4) or P3(0.2)) = MAX (0.4, 0.2) = 0.4
CF(R1) = .7 * .4 = .28
CF(R2) = .3 * .4 = .12

32. Certainty Factor (CF) Formalism
Two properties that are required of the combination operation are:
Commutative – The value should not depend on the order in which the rules are taken.
Asymptotic – The more evidence we have for the belief in a conclusion the higher should be the certainty factor, but if it is not absolutely certain, then it should remain below 1.

33. Certainty Factor (CF) Formalism
Propagation of Certainty Factors
When there are two or more rules supporting the same conclusion CFs are propagated as follows:
CFrevised = CFold + CFnew(1 - CFold) if both CFold and CFnew > 0
= CFold + CFnew(1 + CFold) if both CFold and
CFnew < 0
= otherwise

34. Certainty Factor Example
In a murder trial the defendant is being accused of a first degree murder (hypothesis).The jury must balance the evidences presented by the prosecutor and the defense attorney to decide if the suspect is guilty.
RULE001 IFthe defendant's fingerprints are on the weapon,  
THEN the defendant is guilty. CF=0.75
RULE002 IFthe defendant has a motive,  
THEN the defendant is guilty. CF=0.60
RULE003 IFthe defendant has a alibi,  
THEN he is not guilty. CF=-.80

35. Certainty Factor Example
We start with CF = 0.0 for the defendant being guilty.
After submission of the evidence 1 (fingerprints on the weapon)
CFcomb1 = CF rule1's conclusin * CF evid1
= 0.75 * 0.90 = 0.675
CF revised = CF old + CF new * (1 - CFold)
= (1-0.0) = 0.675

36. Example of CFs Propagation
CFold=0.0
CFnew=0.675
Guilty
CF = 0.0
Guilty
CFrevised=0.675
CFcon1=CFnew=0.675
CFrevised=CFold + CFnew*(1-CFold)
= *(1-0.0)
=0.675
fingerprints
on weapon
CFevid1=0.90
CFrule1=0.75
RULE 1. IF the defendant’s fingerprints are on the weapon
THEN the defendant is guilty
CFcon1=CFevid1*CFrule1 (single premise rule)
=0.9*0.75
=0.675

37. Certainty Factor Example
The defendant’s mother in law says that he had the motive for slaying
CFnew = CFcomb2 = CF rule2's conclusin * CF evid2
= 0.60 * 0.50 = 0.30
CF revised = CF old + CF new * (1 - CFold)
= ( ) =

38. CFrevised=CFold + CFnew*(1-CFold) =0.675 + 0.30*(1-0.675) =0.7725
Guilty
CFrevised=0.772
CFcon2=CFnew=0.30
Guilty
CFrevised=0.675
CFrevised=CFold + CFnew*(1-CFold)
= *( )
=0.7725
Motive exists
CFevid2=0.50
CFrule2=0.60
RULE 2. IF the defendant has a motive
THEN the defendant is guilty of the crime
CFcon2=CFevid2*CFrule2
=0.50*0.60
=0.30
(single premise rule)

39. Certainty Factor Example
A respected judge witnesses for alibi, so a cf of 0.95 is assigned for this evidence
CFcomb3 = CF rule3's conclusin * CF evid3
= 0.95 * (-0.80) = -0.76
CFrevised =
= ( ) / ( ) = 0.052

40. CFold=0.772
CFnew=-0.76
CFcon3=CFnew=-0.76
Guilty
CFrevised=0.772
Guilty
CFrevised=0.052
CFreviced=
Alibi found
CFevid3=0.95
CFrule3= -0.80
= ( )/(1-0.76)
= 0.052
RULE 3. IF the defendant has an alibi
THEN he is not guilty
CFcon3=CFevid3*CFrule3
=0.95*(-0.80)
= -0.76

41. Certainty Factor Example
Confidence Factor in guilty verdict after introduction of all evidences is:

42. Advantages of Certainty Factors
It is a simple computational model that permits experts to estimate their confidence in conclusions being drawn.
It permits the expression of belief and disbelief in each hypothesis, allowing the expression of the effect of multiple sources of evidence.
It allows knowledge to be captured in a rule representation while allowing the quantification of uncertainty.
The gathering of the CF values is significantly easier than the gathering of values for the other methods. No statistical base is required – you merely have to ask the expert for the values.

43. Difficulties Deep Inference Chains
If we have a chain of inference such as:
IF A THEN B CF=0.8
IF B THEN C CF= 0.9
Then because of the multiplication of CFs the resulting CF decreases.
For example if CF(A) = 0.8, then
CF(C) = .8*.8*.9 = .58
With long chain of inferences the final CF may become very small

44. Difficulties Many Rules with same Conclusion
The more rules with the same conclusion the higher the CF value. If there are many rules then CF can become artificially high.

45. Difficulties Conjunctive Rules
If a rule has a number of conjunctive premises, overall CF may be reduced too much.
IF sky dark AND temperature dropping
THEN will rain 0.9
If CF(sky dark) = 1,
CF(temperature dropping) = .1 then
CF(will rain) = min(1, .1)*.9 = .09 whereas if we had
IF the sky dark THEN will rain 0.7
IF temperature dropping THEN will rain 0.5
CF1 = 1 * .7 = 0.7, CF2 = .1 * .5 = 0.05
CF (will rain) = *(1 - .7) = = .715

46. Fuzzy Logic
In everyday speech we use vague or imprecise terms to describe properties.
Fuzzy logic was developed by Zadeh to deal with these imprecise values in a mathematical way.

47. Fuzzy Logic It will allow us to deal with fuzzy rules
IF the temperature is cold
THEN the motor speed stops
IF speed is slow
THEN make acceleration high.

48. Fuzzy Sets
In ordinary set theory, an element from the domain is either in a set or not in a set.
In fuzzy sets, a number in the range 0-1 is attached to an element – the degree to which the element belongs to the set.
A value of 1 means the element is definitely in the set
A value of 0 means the element is definitely not in the set
Other values are grades of membership.
Formally a fuzzy set A from X is given by its membership function which has type
A : X  [0, 1]

49. Fuzzy Sets
Fuzzy set of small men
Small men – Simpler Curve

50. Fuzzy Sets
The following figure shows the representation of three fuzzy sets for small, medium and tall men. We see that a man of height 4.8 feet is considered both small and medium to some degree.

51. Boolean Operations A (x) = 1 - A (x)
The Boolean operations of union, intersection, and complement can be defined in the straightforward manner.
Complement
The operation is
A (x) = 1 - A (x)

52. Boolean Operations AB (x) = min({A (x), B (x)}) Union
Intersection
The intersection of two fuzzy sets A and B is given by
AB (x) = min({A (x), B (x)})
Union
The union of two fuzzy sets A and B is given by
AB (x) = max({A (x), B (x)})

53. Fuzzy Reasoning
In this section, fuzzy rules and how inference is performed on these rules is presented.
This will be illustrated by a fuzzy system used to control an air conditioner. The variables to be used (with fuzzy values) are temperature (of the room) and speed (of the fan motor).

54. Fuzzy Reasoning The rules are given as follows:
IF the temperature is cold
THEN motor speed stops
IF the temperature is cool
THEN motor speed slows
IF the temperature is just right
THEN motor speed medium
IF the temperature is warm
THEN motor speed fast
IF the temperature is hot
THEN motor speed blast
Temperature Fuzzy Sets
Speed Fuzzy Sets

55. Fuzzy Reasoning
In a fuzzy system all the rules fire in parallel, although in the end many will not contribute to the output.
What we need to determine, in the above system is, given a particular value of the temperature how do we calculate the motor speed.

56. Fuzzy Reasoning
Now, the temperature can be measured fairly accurately, but it will lie in several fuzzy sets. For example if the temperature were 17C then from the figure we see that it is about 25% cool and 80% just right.

57. Fuzzy Reasoning
This means that rules 2 and 3 will contribute to the output speed of the motor.
The fuzzy set for the output can be calculated by multiplying the slow graph by .25 and the medium graph by .80 assuming the contribution is proportional to the fuzzy values of the input temperature

58. Fuzzy Reasoning
One way to amalgamate two sets is to sum the values (with a maximum of 1).
Amalgamated sets and average

59. Fuzzy Reasoning
Other ways of amalgamation (e.g. taking maximum) are possible.
Now we need to determine the actual speed of the motor. This can be done by finding the average value of the curve – I.e. the position where the areas on either side of the perpendicular through this point are equal.

60. acknowledgement
Phil Grant: University of Wales Swansea