1. IN THE NAME OF ALLAH DECISION MAKING BY USING THE THEORY OF EVIDENCE STUDENTS: HOSSEIN SHIRZADEH, AHAD OLLAH EZZATI SUPERVISOR: Prof. BAGERI SHOURAKI SPRING 2009 1

2. OUTLINES  INTRODUTION  BELIEF  FRAMES OF DISCERNMENT  COMBINIG THE EVIDENCE  ADVANTAGES OF DS THEORY  DISADVANTAGES OF DS THEORY  BASIC PROBABLITY ASSIGNMENT  BELIEF FUNCTIONS  DEMPSTER RULE OF COMBINATION  ZADEH’S OBJECTION TO DS THEORY  GENERALIZED DS THEORY  AN APPLICATION OF DECISION MAKING METHOD 2

3. INTRODUCTION Introduced by Glenn Shafer in 1976 “ A mathematical theory of evidence ” A new approach to the representation of uncertainty What means uncertainty? Most people don ’ t like uncertainty Applications Expert systems Decision making Image processing, project planning, risk analysis, … 3

4. INTRODUCTION All students of partial belief have tied it to Bayesian theory and I. Committed to the value of idea and defend it II. Rejected the theory (Proof of inviability) 4

5. : Finite set Set of all subsets : Then Bell is called belief function on INTRODUCTION BELIEF FUNCTION 5

6. is called simple support function if There exists a non-empty subset A of and that 6

7. INTRODUCTION THE IDEA OF CHANCE For several centuries the idea of numerical degree of belief has been identified with the idea of chance. Evidence Theory is intelligible only if we reject this unification Chance : A random experiment : unknown outcome The proportion of the time that a particular one the possible outcomes tends to occur 7

8. INTRODUCTION THE IDEA OF CHANCE Chance density – Set of all possible outcomes :X – Chance q(x) specified for each possible outcome – A chance density must satisfy : 8

9. Chance function – Proportion of time that the actual outcome tends to be in a particular subset of X. – Ch is a chance function if and only it obeys the following INTRODUCTION THE IDEA OF CHANCE 9

10. INTRODUCTION CHANCES AS DEGREES OF BELIEF If we know the chances then we will surely adopt them as our degrees of belief We usually don ’ t know the chances – We have little idea about what chance density governs a random experiment – Scientist is interested in a random experiment precisely because it might be governed by any one of several chance densities 10

11. INTRODUCTION CHANCES AS DEGREES OF BELIEF Chances : – Features of the world This is the way shafer addresses chance – Features of our knowledge or belief Simon Laplace – Deterministic Since the advent of Quantum mechanics this view has lost it ’ s grip on physics 11

12. INTRODUCTION BAYESIAN THEORY OF PARTIAL BELIEF Very Popular theory of partial belief – Called Bayesian after Thomas Bayes Adapts the three basic rules for chances as rules for one ’ s degrees of belief based on a given body of evidence. Conditioning : changing one ’ s degree of belief when that evidence is augmented by the knowledge of a particular proposition 12

13. INTRODUCTION BAYESIAN THEORY OF PARTIAL BELIEF obey s When we learn that is true then 13

14. INTRODUCTION BAYESIAN THEORY OF PARTIAL BELIEF The Bayesian theory is contained in Shafer ’ s evidence theory as a restrictive special case. Why is Bayesian Theory too restrictive? – The representation of Ignorance – Combining vs. Conditioning 14

15. INTRODUCTION BAYESIAN THEORY OF PARTIAL BELIEF T HE R EPRESENTATION OF I GNORANCE In Evidence Theory Belief functions – Little evidence: Both the proposition and it ’ s negation have very low degrees of belief – Vacuous belief function 15

16. INTRODUCTION BAYESIAN THEORY OF PARTIAL BELIEF C OMBINATION VS. C ONDITIONING Dempster rule – A method for changing prior opinion in the light of new evidence Deals symmetrically with the new and old evidence Bayesian Theory – Bayes rule of conditioning No Obvious symmetry Must assume exact and full effect of the new evidence is to establish a single proposition with certainty 16

17. In Bayesian Theory: – Can not distinguish between lack of belief and disbelief – can not be low unless is high – Failure to believe A necessitates accordance of belief to – Ignorance represented by : – Important factor in the decline of Bayesian ideas in the nineteenth century – In DS theory INTRODUCTION BAYESIAN THEORY OF PARTIAL BELIEF T HE R EPRESENTATION OF I GNORANCE 17

18. B ELIEF The belief in a particular hypothesis is denoted by a number between 0 and 1 The belief number indicates the degree to which the evidence supports the hypothesis Evidence against a particular hypothesis is considered to be evidence for its negation (i.e., if Θ = {θ 1, θ 2, θ 3 }, evidence against {θ 1 } is considered to be evidence for {θ 2, θ 3 }, and belief will be allotted accordingly) 18

19. F RAMES OF D ISCERNMENT Dempster - Shafer theory assumes a fixed, exhaustive set of mutually exclusive events Θ = {θ 1, θ 2,..., θ n } Same assumption as probability theory Dempster - Shafer theory is concerned with the set of all subsets of Θ, known as the Frame of Discernment 2 Θ = { , {θ 1 }, …, {θ n }, {θ 1, θ 2 }, …, {θ 1, θ 2,... θ n }} Universe of mutually exclusive hypothesis 19

20. F RAMES OF D ISCERNMENT A subset {θ 1, θ 2, θ 3 } implicitly represents the proposition that one of θ 1, θ 2 or θ n is the case The complete set Θ represents the proposition that one of the exhaustive set of events is true So Θ is always true The empty set  represents the proposition that none of the exhaustive set of events is true So  always false 20

21. C OMBINING THE E VIDENCE Dempster-Shafer Theory as a theory of evidence has to account for the combination of different sources of evidence Dempster & Shafer’s Rule of Combination is a essential step in providing such a theory This rule is an intuitive axiom that can best be seen as a heuristic rule rather than a well-grounded axiom. 21

22. A DVANTAGES OF DS THEORY The difficult problem of specifying priors can be avoided In addition to uncertainty, also ignorance can be expressed It is straightforward to express pieces of evidence with different levels of abstraction Dempster’s combination rule can be used to combine pieces of evidence 22

23. D ISADVANTAGES Potential computational complexity problems It lacks a well-established decision theory whereas Bayesian decision theory maximizing expected utility is almost universally accepted. Experimental comparisons between DS theory and probability theory seldom done and rather difficult to do; no clear advantage of DS theory shown. 23

24. B ASIC P ROBABILITY A SSIGNMENT The basic probability assignment (BPA), represented as m, assigns a belief number [0,1] to every member of 2 Θ such that the numbers sum to 1 m (A) represents the maesure of the belief that is committed exactly to A (to individual element A and to no smaller subset) 24

25. B ASIC P ROBABILITY A SSIGNMENT E XAMPLE suppose Diagnostic problem No information 60 of 100 are blue 30 of 100 are blue and rest of them are black or yellow 25

26. B ELIEF F UNCTIONS Obtaining the measure of the total belief committed to A: Belief functions can be characterized without reference to basic probability assignments: 26

27. B ELIEF F UNCTIONS For Θ = {A,B} BPA is unique and can recovered from the belief function 27

28. B ELIEF F UNCTIONS Focal element A subset is a focal element if m(A)>0 Core The union of all the focal elements. Theorem 28

29. B ELIEF F UNCTIONS B ELIEF I NTERVALS Ignorance in DS Theory: The width of the belief interval: The sum of the belief committed to elements that intersect A, but are not subsets of A The width of the interval therefore represents the amount of uncertainty in A, given the evidence 29

30. One ’ s belief about a proposition A are not fully described by one ’ s degree of belief Bel(A) Bel(A) does not reveal to what extend one doubts A Degree of Doubt: Upper probability: The total probability mass that can move into A. B ELIEF F UNCTIONS D EGREES OF D OUBT AND U PPER P ROBABILITIES 30

31. B ELIEF F UNCTIONS D EGREES OF D OUBT AND U PPER P ROBABILITIES E XAMPLE 31 SubsetmBelDouP*P* {}0010 {1}0.1 0.5 {2}0.2 0.40.6 {3}0.1 0.40.6 {1, 2}0.10.40.10.9 {1, 3}0.20.40.20.8 {2, 3}0.20.50.10.9 {1, 2, 3}0.1101 m({1, 2}) = – Bel({1}) – Bel({2}) + Bel({1, 2}) = – 0.1 – 0.2 + 0.4

32. B ELIEF F UNCTIONS B AYESIAN B ELIEF F UNCTIONS A belief function Bel is called Bayesian if Bel is a probability function. The following conditions are equivalent Bel is Bayesian All the focal elements of Bel are singletons For every A ⊆ Θ, The inner measure can be characterized by the condition that the focal elements are pairwise disjoint. 32

33. B ELIEF F UNCTIONS B AYESIAN B ELIEF F UNCTIONS E XAMPLE Suppose SubsetBPABelief Φ00 {a}m1m1 m1m1 {b}m2m2 m2m2 {c}m3m3 m3m3 {a, b}0m 1 + m 2 {a, c}0m 1 + m 3 {b, c}0m 2 + m 3 {a, b, c}0m 1 + m 2 + m 3 = 1 33

34. D EMPSTER R ULE OF C OMBINATION Belief functions adapted to the representation of evidence because they admit a genuine rule of combination. Several belief functions Based on distinct bodies of evidence Computing their “ Orthogonal sum ” using Dempster ’ s rule 34

35. D EMPSTER R ULE OF C OMBINATION C OMBINING T WO B ELIEF F UNCTIONS m 1 : basic probability assignment for Bel 1 A 1,A 2, … A k : Bel 1 ’ s focal elements m 2 : basic probability assignment for Bel 2 B 1,B 2, … B l : Bel 2 ’ s focal elements 35

36. D EMPSTER R ULE OF C OMBINATION C OMBINING T WO B ELIEF F UNCTIONS 36 Probability mass measure of m 1 (A i )m 2 (B j ) committed to

37. D EMPSTER R ULE OF C OMBINATION C OMBINING TWO B ELIEF FUNCTIONS The intersection of two strips m 1 (A i ) and m 2 (B J ) has measure m 1 (A i )m 2 (B J ), since it is committed to both A i and to B J, we say that the joint effect of Bel 1 and Bel 2 is to commit exactly to The total probability mass exactly committed to A: 37

38. D EMPSTER R ULE OF C OMBINATION C OMBINING TWO B ELIEF FUNCTIONS E XAMPLE m 1 ({1}) = 0.3m 1 ({2}) = 0.3m 1 ({1, 2}) = 0.4 m 2 ({1}) = 0.2{1}, 0.06Φ, 0.06{1}, 0.08 m 2 ({2}) = 0.3Φ, 0.09{2}, 0.09{2}, 0.12 m 2 ({1, 2}) = 0.5{1}, 0.15{2}, 0.15{1,2}, 0.2 38 SubsetΦ{1}{2}{1, 2} mcmc 0.150.290.360.2

39. D EMPSTER R ULE OF C OMBINATION C OMBINING TWO B ELIEF FUNCTIONS The only Difficulty some of the squares may be committed to empty set If A i and B j are focal elements of Bel 1 and Bel 2 and if then The only Remedy: Discard all the rectangles committed to empty set Inflate the remaining rectangles by multiplying them with 39

40. D EMPSTER R ULE OF C OMBINATION T HE W EIGHT OF C ONFLICT The renormalizing factor measures the extent of conflict between two belief functions. Every instance in which a rectangle is committed to  corresponds to an instance which Bel 1 and Bel 2 commit probability to disjoint subsets A i and B j 40

41. D EMPSTER R ULE OF C OMBINATION T HE W EIGHT OF C ONFLICT ( CONT.) Bel 1, Bel 2 not conflict at all: k = 0, Con(Bel 1, Bel 2 )= 0 Bel 1, Bel 2 flatly contradict each other: does not exist k = 1, Con(Bel 1, Bel 2 ) = ∞ In previous example k = 0.15 41

42. D EMPSTER ’ S RULE OF COMBINATION Suppose m1 and m2 are basic probability functions over Θ. Then m 1 ⊕ m 2 is given by In previous example 42 SubsetΦ{1}{2}{1, 2} m=m 1 ⊕ m 2 00.34120.42350.2353

43. D EMPSTER R ULE OF C OMBINATION A N A PPLICATION OF DS T HEORY Frame of Discernment: A set of mutually exclusive alternatives: All subsets of FoD form: 43

44. D EMPSTER R ULE OF C OMBINATION A N A PPLICATION OF DS T HEORY Exercise deploys two “evidences in features” m 1 and m 2 m 1 is based on MEAN features from Sensor1 m 1 provides evidences for {SIT} and { ¬ SIT} ({ ¬ SIT} = {STAND, WALK}) m 2 is based on VARIANCE features from Sensor1 m 2 provides evidences for {WALK} and { ¬ WALK } ({ ¬ WALK } = {SIT, STAND}) 44

45. D EMPSTER R ULE OF C OMBINATION A N A PPLICATION OF DS T HEORY 45

46. D EMPSTER R ULE OF C OMBINATION A N A PPLICATION OF DS T HEORY C ALCULATION OF EVIDENCE M 1 z 1 =mean(S1) Evidence Concrete value z 1 (t) Bel(SIT) = 0.2 Pls(SIT) = 1 - Bel( ¬ SIT) = 0.5 m 1Concrete Value (SIT, ¬SIT,  ) = (0.2, 0.5, 0.3) 46

47. D EMPSTER R ULE OF C OMBINATION A N A PPLICATION OF DS T HEORY C ALCULATION OF EVIDENCE M 2 z 2 =variances(S1) Concrete value z 2 (t) Bel(WALK) = 0.4 Pls(WALK) = 1-Bel( ¬ WALK) = 0.5 Evidence m 2Concrete Value (WALK, ¬WALK,  ) = (0.4, 0.5, 0.1) 47

48. D EMPSTER R ULE OF C OMBINATION A N A PPLICATION OF DS T HEORY DS T HEORY C OMBINATION Applying Dempster´s Combination Rule: Due to m({})  Normalization with 0.92 (=1-0.08) m 1 (SIT) = 0.2 m 1 (¬SIT) = m 1 (STAND,WALK) = 0.5 m 1 (ALL) = 0.3 m 2 (WALK) = 0.4m({}) = 0.08m(WALK) = 0.2m(WALK) = 0.12 m 2 (¬WALK) = m 2 (STAND,SIT) = 0.5 m(SIT) = 0.1m(STAND) = 0.25 m(STAND,SIT) = 0.15 m 2 (ALL) = 0.1m(SIT) = 0.02m(STAND,WALK)=0.05m(ALL) = 0.03 48

49. D EMPSTER R ULE OF C OMBINATION A N A PPLICATION OF DS T HEORY N ORMALIZED V ALUES Belief(STAND) = 0.272 Plausibility(STAND) = 1 - (0.108+0.022+0.217+0.13) = 0.523 m 1 (SIT)= 0.2 m 1 (¬SIT) = m 1 (STAND,WALK) =0.5 m 1 (ALL) = 0.3 m 2 (WALK)= 0.40m(WALK)=0.217m(WALK)=0.13 m 2 (¬WALK) = m 2 (STAND,SIT) =0.5 m(SIT)=0.108m(STAND)=0.272 m(STAND,SIT)= 0.163 m 2 (ALL) = 0.1m(SIT)=0.022m(STAND,WALK)=0.054m(ALL)=0.033 49

50. D EMPSTER R ULE OF C OMBINATION A N A PPLICATION OF DS T HEORY B ELIEF AND P LAUSIBILITY (SIT) Ground Truth: 1: Sitting; 2: Standing; 3: Walking 50

51. D EMPSTER R ULE OF C OMBINATION A N A PPLICATION OF DS T HEORY DS C LASSIFICATION 51

52. D EMPSTER R ULE OF C OMBINATION P ROBLEMS IN C OMBINING E VIDENCE Unfortunately, the above approach doesn't work It satisfies the second assumption about mass assignments, that the masses add to 1 But it usually conflicts with the first assumption, that the mass of the empty set is zero Why? Because some subsets X and Y don't intersect, so their intersection is the empty set So when we apply the formula, we end up with non-zero mass assigned to the empty set We can’t arbitrarily assign m 1 ⊕ m 2 (  ) = 0 because the sum of m 1 ⊕ m 2 will no longer be 1 52

53. Z ADEH ’ S O BJECTION TO DS T HEORY Suppose two doctors A and B have the following beliefs about a patient's illness: m A (meningitis) = 0.99 m A (concussion) = 0.00 m A (brain tumor) = 0.01 m B (meningitis) = 0.00 m B (concussion) = 0.99 m B (brain tumor) = 0.01 then k = m A (meningitis) * m B (concussion) + m A (meningitis) * m B (brain tumor) + m A (brain tumor) * m B (concussion) = 0.9999 so mA ⊕ mB (brain tumor) = (.01 *.01) / (1 -.9999) = 1 53

54. G ENERALIZED DS T HEORY Body of evidence Consider Ω = {w 1, w 2,..., w n } {A 1, A 2, …, A n }{m 1, m 2, …, m n }, Φ≠A i Ω Fuzzy Body of evidence Yen’s generalization 54

55. E XAMPLE Consider body of evidence in DS theory over Ω = {1, 2, …, 10} with focal elements: We want to compute Bel(B) and Pls(B) where 55

56. E XAMPLE Acgcording Yen’s generalization A and C whit A and C’s α-cuts then distribute their BPA among α-cuts Coputing belief and plusibility 56

57. A N A PPLICATION OF D ECISION M AKING M ETHOD B ASED ON FUZZIFICATED DS T HEORY A CASE STUDY IN MEDICINE Consider these rules: If “A change in breast skin” Then status is “malignant” If “No change in breast skin” Then status is “unknown” If “Adenoma dwindles” Then status is “benign” If “Adenoma does not dwindle” Then status is “unknown” Suppose we have the following probabilistic P(“A change in breast skin” ) = 0.7 P(“No change in breast skin” ) = 0.3 P(“Adenoma dwindles” ) = 0.4 P(“Adenoma does not dwindle” ) = 0.6 57

58. A N A PPLICATION OF D ECISION M AKING M ETHOD B ASED ON FUZZIFICATED DS T HEORY body of evidence {malignant, [total range]} m 1 (malignant) = 0.7, m 1 ( [total range] ) = 0.3 {benign, [total range]} m 2 (benign) = 0.4, m 2 ( [total range] ) = 0.6 combining body of evidence m 12 (benign) = 0.1476 m 12 (malignant) = 0.5164 m 12 (benign malignant) = 0.1147 m 12 ([total range]) = 0.2213 58

59. A N A PPLICATION OF D ECISION M AKING M ETHOD B ASED ON FUZZIFICATED DS T HEORY Definition Fuzzy Valued Bel and Pls Functions 59

60. A N A PPLICATION OF D ECISION M AKING M ETHOD B ASED ON FUZZIFICATED DS T HEORY Pr(benign) 60 Pr(malignant)

61. A N A PPLICATION OF D ECISION M AKING M ETHOD B ASED ON FUZZIFICATED DS T HEORY Calculating risk functions based on following equation 61

62. A N A PPLICATION OF D ECISION M AKING M ETHOD B ASED ON FUZZIFICATED DS T HEORY a. Fuzzy set of risk function values for benign prediction b.Fuzzy set of risk function values for malignant prediction 62

63. A N A PPLICATION OF D ECISION M AKING M ETHOD B ASED ON FUZZIFICATED DS T HEORY The final step rejecting uncertainties (fuzzyness and ignorance) to obtain a scalar value These answers are calculated by 63

64. THANKS 64