1. Textbook Basics of an Expert System: – “Expert systems: Design and Development,” by: John Durkin, 1994, Chapters 1-4. Uncertainty (Probability, Certainty factor & Fuzzy): – “Expert systems: Design and Development,” by: John Durkin, 1994, Chapters 11-13. – “Artificial intelligence: a guide to intelligent systems,” by: Michael Negnevitsky, 2005, Chapters 3,4. 1

2. Chapter 11 Bayesian approach to inexact reasoning 2

3. What is Uncertainty? Information can be incomplete, inconsistent, uncertain, or all three. Uncertainty: lack of the exact knowledge to reach a perfectly reliable conclusion. Classical logic permits only exact reasoning. IF A is true THEN A is not false IF A is false THEN A is not true 3

4. Sources of uncertain knowledge Weak implications: Need for concrete correlations between IF and THEN parts handling vague associations is required Imprecise language: Our natural language is ambiguous and imprecise. (i.e. often and sometimes, frequently and hardly ever) In 1944, Ray Simpson asked 355 high school and college students to place 20 terms on a scale between 1 and 100. In 1968, Milton Hakel repeated this experiment.

5. Quantification of ambiguous and imprecise terms 5

6. Sources of uncertain knowledge (continued.) Unknown data: When the data is incomplete or missing, the only solution is to accept the value “unknown” and proceed to an approximate reasoning with this value. Combining the views of different experts: Large expert systems usually combine the knowledge and expertise of a number of experts. Unfortunately, experts often have contradictory opinions and produce conflicting rules. To resolve the conflict, attaching a weight to each expert 6

7. Solutions to Handle Uncertainty Probability approach Durkin, Ch.11 & Negnevitsky Ch. 3 Certainty Factor Durkin Ch. 12 & Negnevitsky Ch. 3 Fuzzy Logic (Durkin Ch. 13 & Negnevitsky Ch. 4 7

8. Basic probability theory Probability provides an exact approach for inexact reasoning The probability of an event is the proportion of cases in which the event occurs. A scientific measure of chance. A range between 0 (impossibility) to 1 (certainty). 8

9. Basic probability theory s: success; f :failure throwing a coin, the probability of getting a head s the probability of getting a tail f P(s) = p(f) =0.5

10. Conditional Probability Let A, B be two events in the world A and B are not mutually exclusive (A∩B ≠ null) The probability that event A will occur if event B occurs is called the conditional probability. p(A|B): Conditional probability of event A occurring given that event B has occurred. 10

11. Conditional Probability P(A∩B): Joint Probability Similarly, Hence, 11

12. Bayesian Rule where: p(A|B) is the conditional probability that event A occurs given that event B has occurred; p(B|A) is the conditional probability of event B occurring given that event A has occurred; p(A) is the probability of event A occurring; p(B) is the probability of event B occurring. 12

13. Bayesian Rule 13

14. Example: Chest Pain 14

15. Joint Probability on a number of mutually exclusive and exhaustive events 15

16. Bayesian rule with mutually exclusive and exhaustive events Replacing in Bayesian rule 16

17. Bayesian Reasoning Suppose all rules in the knowledge base are represented in the following form: If E is true, then H is true {with probability p} This rule implies that if event E occurs, then the probability that event H will occur is p. H: hypothesis E: evidence 17

18. The Bayesian rule in terms of hypotheses and evidence 18

19. The Bayesian rule in expert systems 19

20. The Bayesian rule with multiple hypothesis and multiple evidences 20

21. The Bayesian rule with multiple hypothesis and multiple evidences assumption: conditional independency among different evidences 21

22. Example: Ranking potentially true hypothesis Suppose an expert, given three conditionally independent evidences E1, E2 and E3, creates three mutually exclusive and exhaustive hypotheses H1, H2 and H3. 22 H1: Cold H2: Allergy H3: Flu E1: Cough E2: Fever E3: runny nose

23. Example: Ranking potentially true hypothesis Assume that we first observe evidence E3. The expert system computes the posterior probabilities for all hypotheses as After evidence E3 is observed, belief in hypothesis H1 decreases and becomes equal to belief in hypothesis H2. Belief in hypothesis H3 increases and even nearly reaches beliefs in hypotheses H1 and H2. 23 H1: Cold H2: Allergy H3: Flu E1: Cough E2: Fever E3: runny nose

24. Example: Ranking potentially true hypothesis Suppose now that we observe evidence E1. The posterior probabilities are calculated as Hypothesis H2 has now become the most likely one. 24 H1: Cold H2: Allergy H3: Flu E1: Cough E2: Fever E3: runny nose

25. Example: Ranking potentially true hypothesis After observing evidence E2, the final posterior probabilities for all hypotheses are calculated: Although the initial ranking was H1, H2 and H3, only hypotheses H1 and H3 remain under consideration after all evidences (E1, E2 and E3) were observed. 25

26. Uncertain Evidence (LS & LN) 26 If E {LS, LN}, then H LS (likelihood of sufficiency): represents a measure of the expert belief in hypothesis H if evidence E is present. LN (likelihood of necessity): a measure of discredit to hypothesis H, if evidence E is missing.

27. Uncertain Evidence (LS & LN) Note that LN cannot be derived from LS. The domain expert must provide both values independently High values of LS (LS>>1) indicate that the rule strongly supports the hypothesis if the evidence is observed, and low values of LN (0<LN<1) suggest that the rule also strongly opposes the hypothesis if the evidence is missing. 27 Review

28. Uncertain Evidence (LS & LN) Usually (but not always): Example: FORCAST expert system of London: 28 Review

29. Effects of LS and LN on Hypothesis 29 Review

30. How to calculate posterior probabilities from LS & LN 30 Review Q: Prove that the above equations are the same Bayesian rule.

31. Example: the weather forecasting 31

32. Example: the weather forecasting (non-probabilistic solution) 32 Simple non-probabilistic knowledge base: Rule: 1 IF today is rain, THEN tomorrow is rain Rule: 2 IF today is dry, THEN tomorrow is dry Using these rules we will make only ten mistakes

33. Example: the weather forecasting (probabilistic solution) 33 Simple Probabilistic knowledge base:

34. Example: the weather forecasting p(tomorrow is rain | today is rain) 34

35. Example: the weather forecasting Complete knowledge base 35

36. Assignment 3, (Durkin, ch. 11) Due Date: 93/8/29 36