1. KETIDAKPASTIAN (UNCERTAINTY) Yeni Herdiyeni – http://ilkom.fmipa.ipb.ac.id/yeni “Uncertainty is defined as the lack of the exact knowledge that would enable us to reach a perfectly reliable conclusion”

2. Topik  Pengertian Ketidakpastian (Uncertainty)  Konsep Dasar Peluang (Probability)  Faktor kepastian (Certainty Factor)  Penalaran Bayes (Bayesian Reasoning)  Fuzzy  Neural Network  Genetic Algorithm

3. Ketidakpastian Hujan atau Tidak ya….???

4. Expert System in Nursing  For example, a nursing diagnosis may require to know if a patient had suffered from a certain allergy?  The patient may not know or remember the answer to this question. Therefore, the patient information is unavailable to ensure a correct diagnosis.  This means that decisions will often be based on incomplete or uncertain data. Clearly, this may result in uncertain conclusions.

5. Ide  Konsep Dasar Peluang (Probability)  Faktor kepastian (Certainty Factor)  Penalaran Bayes (Bayesian Reasoning)  Fuzzy  Neural Network  Genetic Algorithm

6. Aplikasi  Financial Market : Stock Market  Games : Gambling  Weather Forecasts  Risk Management  dll

7. Studi Kasus  PASS : An Expert System with Certainty Factors for Predicting Student Success  Variabel : Jenis Kelamin, spesialisasi, Peringkat  Peringkat:  Moderate : >=10 and <12.5  Good : >=12.5 and <15.5  Very Good: >=15.5 and <18.5  Excellent: >=18.5 and <=20

8. Ketidakpastian  Informasi seringkali tidak lengkap, tidak konsisten, tidak pasti (uncertain)  Weak Implication  Imprecise language : ambigu, imprecise  Unknown data  Menggabungkan perbendaan pandangan dari pakar

9. 9 Sources of Uncertain Knowledge Imprecise language

10. Uncertainty  In evidence:  'if pressure gauge is high then liquid is boiling' Electrical components are notoriously faulty, the pressure gauge may actually be stuck  In inferring conclusion:  'if patient has a sore throat then patient has tonsillitis' A doctor may infer such a conclusion but it would not be an absolute, binary, one.  Vagueness of language:  'if the car is a Porsch then it is fast'. What do we mean when using the term fast?

11. Representasi Ketidakpastian  Certainty Theory  Probabilistic  Theory of evidence (Bayes Theorm)  Fuzzy logic  Neural Network  GA

12. Certainty Theory

13.  A certainty factor (cf ), a number to measure the expert’s belief.  Maximum value of the certainty factor is +1.0 (definitely true) and minimum -1.0 (definitely false).  Two aspects:  Certainty in the Evidence The evidence can have a certainty factor attached  Certainty in Rule  Note CF values are not probabilities, but informal measures of confidence

14. Certainty Factors TermCertainty Factor Definitely not Almost certainly not-0.8 Probably not-0.6 Maybe not-0.4 Unknown-0.2 to +0.2 Maybe+0.4 Probably+0.6 Almost certainly+0.8 Definitely+1.0  Uncertain term and their intepretation

15. Expert Systems with Certainty Theory  In expert systems with certainty factors, the knowledge base consists of a set of rules that have the following syntax: IF THEN {cf }  Where cf represents belief in hypothesis H given that evidence E has occurred.

16.  The certainty factor assigned by a rule is propagated through the reasoning chain  Now belief in hypothesis H given that evidence E has occurred, is:  cf (H,E) = cf (E) * cf(Rule)  For example: IF sky is clear THEN the forecast is sunny {cf 0.8}  and current certainty factor of “sky is clear” is 0.5  cf (H,E) = 0.5 * 0.8 = 0.4 Example The certainty that the forcast is sunny = 0.4

17. Multiple Antecedents  So if a certainty is attached to the evidence  What happens when we have rules with more than one piece of evidence?  With conjunctions (i.e. and)  Use the minimum cf of evidence  With disjunctions (i.e. or)  Use the maximum cf of evidence

18. Conjunctive Antecedents - Example  Conjunctive Rules: cf(H, E 1 and E 2 and …E i ) = min{c(E 1, E 2, …E i )} *cf(Rule) IF there are dark clouds E 1 AND the wind is strongerE 2 THEN it will rain {cf 0.8}  So assume that cf (E 1 ) = 0.5 and cf (E 2 ) = 0.9, then cf(H, E) = min{0.5, 0.9} * 0.8 = 0.4

19. Disjunctive Antecedents - Example  Disjunctive Rules: cf(H, E 1 or E 2 or …E i ) = max{cf(E 1, E 2, …E i )} * cf(Rule) IF there are dark clouds E 1 OR the wind is strongerE 2 THEN it will rain {cf 0.8}  Again assume that cf(E 1 ) = 0.5 and cf(E 2 ) = 0.9, then cf(H, E) = max{0.5, 0.9} * 0.8 = 0.72

20.  What if two pieces of evidence leads to the same conclusion?  Common sense suggests that, if two pieces of evidence from different sources support the same hypothesis then confidence in this hypothesis should increase more than if only one piece of evidence had been obtained. Similarly Concluded Rules Rule 1: IF A is X THEN C is Z { cf 0.8} Rule 2:IF B is Y THEN C is Z { cf 0.6}

21. Similarly Concluded Rules - Example  Two rules can lead to same conclusion: IF weatherperson predicts rain E 1 THEN it will rain { cf 1 0.7} IF farmer predicts rain E 2 THEN it will rain { cf 2 0.9} Assume cf(E 1 ) = 1.0 and that cf(E 2 ) = 1.0 cfcfcf cf 1 (H 1, E 1 ) = cf (E 1 ) * cf (Rule 1 ) cf cf 1 (H 1, E 1 ) = 1.0 * 0.7 = 0.7 cfcfcf cf 2 (H 2, E 2 ) = cf (E 2 ) * cf (Rule 2 ) cf cf 2 (H 2, E 2 ) = 1.0 * 0.9 = 0.9

22. Similarly Concluded Rules  If we obtain supporting evidence for hypothesis from two sources, then should feel more confident in conclusion  So: cf combine ( cf 1, cf 2 ) = cf 1 + cf 2 * (1 - cf 1 ) = 0.7 + 0.9 * ( 1 – 0.7) = 0.97

23. Certainty Theory  So certainty theory allows us to reason with uncertainty in:  evidence  inferring conclusion – i.e. the hypothesis  But what about vagueness of language:  'if the car is a Porsch then it is fast'.  For this we use Fuzzy Logic

24. Probability

25. 21 st September 2006Bogdan L. Vrusias © 2006 25 Basic Probability Theory  The concept of probability has a long history that goes back thousands of years when words like “probably”, “likely”, “maybe”, “perhaps” and “possibly” were introduced into spoken languages. However, the mathematical theory of probability was formulated only in the 17th century.  The probability of an event is the proportion of cases in which the event occurs. Probability can also be defined as a scientific measure of chance.

26. 26 Basic Probability Theory  Probability can be expressed mathematically as a numerical index with a range between zero (an absolute impossibility) to unity (an absolute certainty).  Most events have a probability index strictly between 0 and 1, which means that each event has at least two possible outcomes: favourable outcome or success, and unfavourable outcome or failure.

27. 27 Basic Probability Theory  If s is the number of times success can occur, and f is the number of times failure can occur, then  and p + q = 1  If we throw a coin, the probability of getting a head will be equal to the probability of getting a tail. In a single throw, s = f = 1, and therefore the probability of getting a head (or a tail) is 0.5.

28. The Axioms of Probability  0 <= P(A) <= 1  P(True) = 1  P(False) = 0  P(A or B) = P(A) + P(B) - P(A and B)

29. Theorems from the Axioms  0 <= P(A) <= 1, P(True) = 1, P(False) = 0  P(A or B) = P(A) + P(B) - P(A and B) From these we can prove: P(not A) = P(~A) = 1-P(A) A B P(A or B) B P(A and B)

30. Copyright © Andrew W. Moore Another important theorem  0 <= P(A) <= 1, P(True) = 1, P(False) = 0  P(A or B) = P(A) + P(B) - P(A and B) From these we can prove: P(A) = P(A ^ B) + P(A ^ ~B)  How? A B P(A or B) B P(A and B)

31. 31 Conditional Probability  Let A be an event in the world and B be another event.  Suppose that events A and B are not mutually exclusive, but occur conditionally on the occurrence of the other.  The probability that event A will occur if event B occurs is called the conditional probability.  Conditional probability is denoted mathematically as p(A|B) in which the vertical bar represents "given" and the complete probability expression is interpreted as  “Conditional probability of event A occurring given that event B has occurred”.

32. Copyright © Andrew W. Moore Conditional Probability  P(A|B) = Fraction of worlds in which B is true that also have A true F H H = “Have a headache” F = “Coming down with Flu” P(H) = 1/10 P(F) = 1/40 P(H|F) = 1/2 “Headaches are rare and flu is rarer, but if you’re coming down with ‘flu there’s a 50-50 chance you’ll have a headache.”

33. Copyright © Andrew W. Moore Conditional Probability F H H = “Have a headache” F = “Coming down with Flu” P(H) = 1/10 P(F) = 1/40 P(H|F) = 1/2 P(H|F) = Fraction of flu-inflicted worlds in which you have a headache = #worlds with flu and headache ------------------------------------ #worlds with flu = Area of “H and F” region ------------------------------ Area of “F” region = P(H ^ F) ----------- P(F)

34. Copyright © Andrew W. Moore Definition of Conditional Probability P(A ^ B) P(A|B) = ----------- P(B) Corollary: The Chain Rule P(A ^ B) = P(A|B) P(B)

35. Copyright © Andrew W. Moore Probabilistic Inference F H H = “Have a headache” F = “Coming down with Flu” P(H) = 1/10 P(F) = 1/40 P(H|F) = 1/2 One day you wake up with a headache. You think: “Drat! 50% of flus are associated with headaches so I must have a 50-50 chance of coming down with flu” Is this reasoning good?

36. Copyright © Andrew W. Moore Probabilistic Inference F H H = “Have a headache” F = “Coming down with Flu” P(H) = 1/10 P(F) = 1/40 P(H|F) = 1/2 P(F ^ H) = … P(F|H) = …

37. 21 st September 2006Bogdan L. Vrusias © 2006 37 Conditional Probability  The number of times A and B can occur, or the probability that both A and B will occur, is called the joint probability of A and B. It is represented mathematically as p(A  B). The number of ways B can occur is the probability of B, p(B), and thus  Similarly, the conditional probability of event B occurring given that event A has occurred equals

38. 38 Conditional Probability Hence And Substituting the last equation into the equation yields the Bayesian rule:

39. Copyright © Andrew W. Moore What we just did… P(A ^ B) P(A|B) P(B) P(B|A) = ----------- = --------------- P(A) P(A) This is Bayes Rule Bayes, Thomas (1763) An essay towards solving a problem in the doctrine of chances. Philosophical Transactions of the Royal Society of London, 53:370-418

40. Copyright © Andrew W. Moore Another way to understand the intuition Thanks to Jahanzeb Sherwani for contributing this explanation:

41. Papers  Uncertainty Management in Expert System  PASS : An Expert System with Certainty Factors for Predicting Student Success  FuzzyCLIPS

42. Next Week  Bayesian Inference  Fuzzy