1. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 1 CPE/CSC 481: Knowledge-Based Systems Dr. Franz J. Kurfess Computer Science Department Cal Poly

2. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 2 Usage of the Slides u these slides are intended for the students of my CPE/CSC 481 “Knowledge-Based Systems” class at Cal Poly SLO u if you want to use them outside of my class, please let me know (fkurfess@calpoly.edu)fkurfess@calpoly.edu u I usually put together a subset for each quarter as a “Custom Show” u to view these, go to “Slide Show => Custom Shows”, select the respective quarter, and click on “Show” u To print them, I suggest to use the “Handout” option u 4, 6, or 9 per page works fine u Black & White should be fine; there are few diagrams where color is important

3. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 3 Overview Reasoning and Uncertainty u Motivation u Objectives u Sources of Uncertainty and Inexactness in Reasoning u Incorrect and Incomplete Knowledge u Ambiguities u Belief and Ignorance u Probability Theory u Bayesian Networks u Certainty Factors u Belief and Disbelief u Dempster-Shafer Theory u Evidential Reasoning u Important Concepts and Terms u Chapter Summary

4. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 4 Logistics u Introductions u Course Materials u textbooks (see below) u lecture notes u PowerPoint Slides will be available on my Web page u handouts u Web page u http://www.csc.calpoly.edu/~fkurfess http://www.csc.calpoly.edu/~fkurfess u Term Project u Lab and Homework Assignments u Exams u Grading

5. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 5 Bridge-In

6. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 6 Pre-Test

7. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 7 Motivation u reasoning for real-world problems involves missing knowledge, inexact knowledge, inconsistent facts or rules, and other sources of uncertainty u while traditional logic in principle is capable of capturing and expressing these aspects, it is not very intuitive or practical u explicit introduction of predicates or functions u many expert systems have mechanisms to deal with uncertainty u sometimes introduced as ad-hoc measures, lacking a sound foundation

8. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 8 Objectives u be familiar with various sources of uncertainty and imprecision in knowledge representation and reasoning u understand the main approaches to dealing with uncertainty u probability theory v Bayesian networks v Dempster-Shafer theory u important characteristics of the approaches v differences between methods, advantages, disadvantages, performance, typical scenarios u evaluate the suitability of those approaches u application of methods to scenarios or tasks u apply selected approaches to simple problems

9. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 9 Evaluation Criteria

10. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 10 Introduction u reasoning under uncertainty and with inexact knowledge u frequently necessary for real-world problems u heuristics u ways to mimic heuristic knowledge processing u methods used by experts u empirical associations u experiential reasoning u based on limited observations u probabilities u objective (frequency counting) u subjective (human experience ) u reproducibility u will observations deliver the same results when repeated

11. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 11 Dealing with Uncertainty u expressiveness u can concepts used by humans be represented adequately? u can the confidence of experts in their decisions be expressed? u comprehensibility u representation of uncertainty u utilization in reasoning methods u correctness u probabilities v adherence to the formal aspects of probability theory u relevance ranking v probabilities don’t add up to 1, but the “most likely” result is sufficient u long inference chains v tend to result in extreme (0,1) or not very useful (0.5) results u computational complexity u feasibility of calculations for practical purposes

12. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 12 Sources of Uncertainty u data u data missing, unreliable, ambiguous, u representation imprecise, inconsistent, subjective, derived from defaults, … u expert knowledge u inconsistency between different experts u plausibility v “best guess” of experts u quality v causal knowledge v deep understanding v statistical associations v observations u scope v only current domain, or more general

13. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 13 Sources of Uncertainty (cont.) u knowledge representation u restricted model of the real system u limited expressiveness of the representation mechanism u inference process u deductive v the derived result is formally correct, but inappropriate v derivation of the result may take very long u inductive v new conclusions are not well-founded v not enough samples v samples are not representative u unsound reasoning methods v induction, non-monotonic, default reasoning

14. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 14 Uncertainty in Individual Rules u errors u domain errors u representation errors u inappropriate application of the rule u likelihood of evidence u for each premise u for the conclusion u combination of evidence from multiple premises

15. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 15 Uncertainty and Multiple Rules u conflict resolution u if multiple rules are applicable, which one is selected v explicit priorities, provided by domain experts v implicit priorities derived from rule properties v specificity of patterns, ordering of patterns creation time of rules, most recent usage, … u compatibility u contradictions between rules u subsumption v one rule is a more general version of another one u redundancy u missing rules u data fusion v integration of data from multiple sources

16. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 16 Basics of Probability Theory u mathematical approach for processing uncertain information u sample space set X = {x 1, x 2, …, x n } u collection of all possible events u can be discrete or continuous u probability number P(x i ) reflects the likelihood of an event x i to occur u non-negative value in [0,1] u total probability of the sample space (sum of probabilities) is 1 u for mutually exclusive events, the probability for at least one of them is the sum of their individual probabilities u experimental probability v based on the frequency of events u subjective probability v based on expert assessment

17. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 17 Compound Probabilities u describes independent events u do not affect each other in any way u joint probability of two independent events A and B P(A  B) = n(A  B) / n(s) = P(A) * P (B) where n(S) is the number of elements in S u union probability of two independent events A and B P(A  B) = P(A) + P(B) - P(A  B) = P(A) + P(B) - P(A) * P (B)

18. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 18 Conditional Probabilities u describes dependent events u affect each other in some way u conditional probability of event A given that event B has already occurred P(A|B) = P(A  B) / P(B)

19. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 19 Advantages and Problems: Probabilities u advantages u formal foundation u reflection of reality (a posteriori) u problems u may be inappropriate v the future is not always similar to the past u inexact or incorrect v especially for subjective probabilities u ignorance v probabilities must be assigned even if no information is available v assigns an equal amount of probability to all such items u non-local reasoning v requires the consideration of all available evidence, not only from the rules currently under consideration u no compositionality v complex statements with conditional dependencies can not be decomposed into independent parts

20. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 20 Bayesian Approaches u derive the probability of a cause given a symptom u has gained importance recently due to advances in efficiency u more computational power available u better methods u especially useful in diagnostic systems u medicine, computer help systems u inverse or a posteriori probability u inverse to conditional probability of an earlier event given that a later one occurred

21. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 21 Bayes’ Rule for Single Event u single hypothesis H, single event E P(H|E) = (P(E|H) * P(H)) / P(E) or u P(H|E) = (P(E|H) * P(H) / (P(E|H) * P(H) + P(E|  H) * P(  H) )

22. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 22 Bayes’ Rule for Multiple Events u multiple hypotheses H i, multiple events E 1, …, E n P(H i |E 1, E 2, …, E n ) = (P(E 1, E 2, …, E n |H i ) * P(H i )) / P(E 1, E 2, …, E n ) or P(H i |E 1, E 2, …, E n ) = (P(E 1 |H i ) * P(E 2 |H i ) * …* P(E n |H i ) * P(H i )) /  k P(E 1 |H k ) * P(E 2 |H k ) * … * P(E n |H k )* P(H k ) with independent pieces of evidence E i

23. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 23 Using Bayesian Reasoning: Spam Filters

24. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 24 Advantages and Problems of Bayesian Reasoning u advantages u sound theoretical foundation u well-defined semantics for decision making u problems u requires large amounts of probability data v sufficient sample sizes u subjective evidence may not be reliable u independence of evidences assumption often not valid u relationship between hypothesis and evidence is reduced to a number u explanations for the user difficult u high computational overhead

25. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 25 Certainty Factors u denotes the belief in a hypothesis H given that some pieces of evidence E are observed u no statements about the belief means that no evidence is present u in contrast to probabilities, Bayes’ method u works reasonably well with partial evidence u separation of belief, disbelief, ignorance u shares some foundations with Dempster-Shafer (DS) theory, but is more practical u introduced in an ad-hoc way in MYCIN u later mapped to DS theory

26. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 26 Belief and Disbelief u measure of belief u degree to which hypothesis H is supported by evidence E u MB(H,E) = 1 if P(H) =1 (P(H|E) - P(H)) / (1- P(H)) otherwise u measure of disbelief u degree to which doubt in hypothesis H is supported by evidence E u MD(H,E) = 1 if P(H) =0 (P(H) - P(H|E)) / P(H)) otherwise

27. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 27 Certainty Factor u certainty factor CF u ranges between -1 (denial of the hypothesis H) and +1 (confirmation of H) u allows the ranking of hypotheses u difference between belief and disbelief CF (H,E) = MB(H,E) - MD (H,E) u combining antecedent evidence u use of premises with less than absolute confidence v E 1  E 2 = min(CF(H, E 1 ), CF(H, E 2 )) v E 1  E 2 = max(CF(H, E 1 ), CF(H, E 2 )) v  E =  CF(H, E)

28. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 28 Combining Certainty Factors u certainty factors that support the same conclusion u several rules can lead to the same conclusion u applied incrementally as new evidence becomes available CF rev (CF old, CF new ) = CF old + CF new (1 - CF old ) if both > 0 CF old + CF new (1 + CF old ) if both < 0 CF old + CF new / (1 - min(|CF old |, |CF new |)) if one < 0

29. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 29 Characteristics of Certainty Factors AspectProbabilityMBMDCF Certainly trueP(H|E) = 1101 Certainly false P(  H|E) = 1 01 No evidenceP(H|E) = P(H)000 Ranges measure of belief 0 ≤ MB ≤ 1 measure of disbelief 0 ≤ MD ≤ 1 certainty factor -1 ≤ CF ≤ +1

30. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 30 Advantages and Problems of Certainty Factors u Advantages u simple implementation u reasonable modeling of human experts’ belief v expression of belief and disbelief u successful applications for certain problem classes u evidence relatively easy to gather v no statistical base required u Problems u partially ad hoc approach v theoretical foundation through Dempster-Shafer theory was developed later u combination of non-independent evidence unsatisfactory u new knowledge may require changes in the certainty factors of existing knowledge u certainty factors can become the opposite of conditional probabilities for certain cases u not suitable for long inference chains

31. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 31 Dempster-Shafer Theory u mathematical theory of evidence u uncertainty is modeled through a range of probabilities v instead of a single number indicating a probability u sound theoretical foundation u allows distinction between belief, disbelief, ignorance (non- belief) u certainty factors are a special case of DS theory

32. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 32 DS Theory Notation u environment  = {O 1, O 2,..., O n } u set of objects O i that are of interest u  = {O 1, O 2,..., O n } u frame of discernment FD u an environment whose elements may be possible answers u only one answer is the correct one u mass probability function m u assigns a value from [0,1] to every item in the frame of discernment u describes the degree of belief in analogy to the mass of a physical object u mass probability m(A) u portion of the total mass probability that is assigned to a specific element A of FD

33. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 33 Belief and Certainty u belief Bel(A) in a set A u sum of the mass probabilities of all the proper subsets of A v all the mass that supports A u likelihood that one of its members is the conclusion u also called support function u plausibility Pls(A) u maximum belief of A u upper bound for the range of belief u certainty Cer(A) u interval [Bel(A), Pls(A)] v also called evidential interval u expresses the range of belief

34. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 34 Combination of Mass Probabilities u combining two masses in such a way that the new mass represents a consensus of the contributing pieces of evidence u set intersection puts the emphasis on common elements of evidence, rather than conflicting evidence u m 1  m 2 (C)=  X  Y m 1 (X) * m 2 (Y) =C m 1 (X) * m 2 (Y) / (1-  X  Y) =C m 1 (X) * m 2 (Y) where X, Y are hypothesis subsets and C is their intersection C = X  Y  is the orthogonal or direct sum

35. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 35 Differences Probabilities - DF Theory Aspect ProbabilitiesDempster-Shafer Aggregate Sum  i Pi = 1m(  ) ≤ 1 Subset X  Y P(X) ≤ P(Y)m(X) > m(Y) allowed relationship X,  X (ignorance) P(X) + P (  X) = 1m(X) + m(  X) ≤ 1

36. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 36 Evidential Reasoning u extension of DS theory that deals with uncertain, imprecise, and possibly inaccurate knowledge u also uses evidential intervals to express the confidence in a statement u lower bound is called support (Spt) in evidential reasoning, and belief (Bel) in Dempster-Shafer theory u upper bound is plausibility (Pls)

37. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 37 Evidential Intervals MeaningEvidential Interval Completely true[1,1] Completely false[0,0] Completely ignorant[0,1] Tends to support[Bel,1] where 0 < Bel < 1 Tends to refute[0,Pls] where 0 < Pls < 1 Tends to both support and refute[Bel,Pls] where 0 < Bel ≤ Pls< 1 Bel: belief; lower bound of the evidential interval Pls: plausibility; upper bound

38. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 38 Advantages and Problems of Dempster-Shafer u advantages u clear, rigorous foundation u ability to express confidence through intervals v certainty about certainty u proper treatment of ignorance u problems u non-intuitive determination of mass probability u very high computational overhead u may produce counterintuitive results due to normalization u usability somewhat unclear

39. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 39 Post-Test

40. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 40 Evaluation u Criteria

41. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 41 Important Concepts and Terms u Bayesian networks u belief u certainty factor u compound probability u conditional probability u Dempster-Shafer theory u disbelief u evidential reasoning u inference u inference mechanism u ignorance u knowledge u knowledge representation u mass function u probability u reasoning u rule u sample u set u uncertainty

42. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 42 Summary Reasoning and Uncertainty u many practical tasks require reasoning under uncertainty u missing, inexact, inconsistent knowledge u variations of probability theory are often combined with rule-based approaches u works reasonably well for many practical problems u Bayesian networks have gained some prominence u improved methods, sufficient computational power

43. © 2002-9 Franz J. Kurfess Reasoning under Uncertainty 43