BAYESIAN NETWORKS AND TRUST FUSION WITH SUBJECTIVE LOGIC

Transcription

1 BYESIN NETWORKS ND TRUST FUSION WITH SUBJECTIVE LOGIC Fusion 2016, Heidelberg udun Jøsang, Universit of Oslo 1

2 bout me Prof. udun Jøsang, Universit of Oslo, 2008 Research interests Information Securit Baesian Reasoning Bio MSc Telecom, NTH, 1988 Engineer, lcatel Telecom MSc Securit, London 1993 PhD Securit, NTNU 1998.Prof. QUT, ustralia, 2000 Department of Universit of Oslo 2

3 Tutorial overview 1. Representations of subjective opinions 1. Decision making with subjective opinions 2. Operators of subjective logic 1. pplications of subjective logic: Trust fusion and transitivit Trust networks Baesian reasoning Subjective networks 3

4 The General Idea of Subjective Logic Logic Probabilit Probabilistic Logic Uncertaint & Subjectivit Subjective Logic 4

5 5 Probabilistic Logic Eamples Binar Logic Probabilistic logic ND: p = pp OR: p = p + p pp MP: {, } MT: {, } p a p a p a p + = p a p a p a p + = p p p p p + = p p p p p + = a: base rate

6 Probabilit and Uncertaint Frequentist aleator: Confident when based on much observation evidence Unconfident when based on little observation evidence E.g.: Probabilit of heads when flipping coin is ½ and confident Subjective epistemic: Confident when dnamics of situation are known Unconfident when dnamics of situation are unknown E.g. Probabilit of Oswald killed Kenned is ½ but unconfident 6

7 Domains, variables and opinions Binar domain = {, } Binar variable = Binomial opinion 3-ar domain Random variable Multinomial opinion Hperdomain R Hpervariable R Hpernomial opinion R

8 Domains and Hperdomains domain is a state space of distinct possibilities Powerset P = 2, set of subsets, including {, } Reduced powerset R = P \ {, } Hperdomain R = { 1, 2, 3, 4, 5, 6 } C called Composite set C = { 4, 5, 6 } Cardinalities = 3 in this eample P = 2 R = 8 in this eample R = 2 2 = 6 in this eample

9 Binomial Opinion Binar domain Binar variable = Multinomial Opinion n-ar domain Random variable Hpernomial Opinion hperdomain R Hpervariable R R Domain Opinion representation PDF representation Beta PDF over Dirichlet PDF over Hper Dirichlet over 9

10 Binomial subjective opinions Belief mass and base rate on binar domains b = b is observer s belief in d = b is observer s disbelief in u = b is observer s uncertaint about a is the base rate of Binomial opinion ω = b, d, u, a base rate of Binar domain b + d + u =1 10

11 What are base rates? In probabilit and statistics, base rate refers to categor probabilit unconditioned on evidence, often referred to as prior probabilities. For eample, if it were the case that 1% of the public are "medical professionals" and 99% of the public are not "medical professionals, then the base rates in this case are 1% and 99%, respectivel. E.g. when picking a random person, the prior probabilit of being a medical professional is 1% 11

12 Barcentric representation of binomial opinions Ordered quadruple: ω = b, d, u, a b : belief d : disbelief u : uncertaint vacuit of evidence a : base rate Point defined b additivit: b + d + u =1 Projected probabilit: P verte disbelief = b + a u u verte uncertaint b ω u d P a verte belief Eample ω =0.4, 0.2, 0.4, 0.9, P =

13 Opinion tpes bsolute opinion: b =1. Equivalent to TRUE. Dogmatic opinion: u =0. Equivalent to probabilities. Vacuous opinion: u =1. Equivalent to UNDEFINED. General uncertain opinion: u 0. 13

14 Beta PDF representation Γ α + β Beta p, α, β = Γ α Γ β p α 1 1 p β 1 α = r + Wa β = s + W1-a r: # observations of s: # observations of a: base rate of W = 2: non-informative prior weight E: Epected probabilit E = P Betap Beta Probabilit Densit Function E p Eample: r = 2, s = 1, a = 0.9, E=

15 15 Binomial Opinion Beta PDF r,s,a represents Beta PDF evidence parameters. b,d,u,a represents binomial opinion. P = E Op Beta: Beta Op: = + + = = 1 / / u d b u Wd s u Wb r = = = W s r W W s r s W s r r u d b W = 2

16 Online demo 16

17 Likelihood and Confidence Likelihood levels: bsolutel not Ver unlikel Unlikel Somewhat unlikel Chances about even Somewhat likel Likel Ver likel bsolutel Confidence levels: No confidence E 9E 8E 7E 6E 5E 4E 3E 2E 1E Low confidence D 9D 8D 7D 6D 5D 4D 3D 2D 1D Some confidence C 9C 8C 7C 6C 5C 4C 3C 2C 1C High confidence B 9B 8B 7B 6B 5B 4B 3B 2B 1B Total confidence

18 Soliciting opinions from people People find it difficult to epress opinions as numerical values Qualitative verbal categories are intuitivel easier Opinions have 2-dimensional qualit Likelihood dimension Confidence dimension i.e. certaint Suitable categories depend on application / contet Eample shows 9 likelihood and 5 confidence levels 1 corresponds to TRUE 9 corresponds to FLSE Low confidence is most natural with medium likelihood 18

19 Mapping qualitative to opinion Categories mapped to corresponding field of triangle Mapping depends on base rate Non-eistent categories depending on base-rates 9E 8E 7E 6E 5E 4E 3E 2E 1E 9E 8E 7E 6E 5E 4E 3E 2E 1E 9D 8D 7D 6D 5D 4D 3D 2D 1D 9D 8D 7D 6D 5D 4D 3D 2D 1D 9C 8C 7C 6C 5C 4C 3C 2C 1C 9C 8C 7C 6C 5C 4C 3C 2C 1C 9B 8B 7B 6B 5B 4B 3B 2B 1B B 8B 7B 6B 5B 4B 3B 2B 1B base rate a = 1/3 base rate a = 2/3 19

20 Mapping categories to opinions Overla categor matri with opinion triangle Matri skewed as a function of base rate Not all categories map to opinions For a low base rate, it is impossible to describe an event as highl likel and uncertain, but possible to describe it as highl unlikel and uncertain. E.g. with regard to tuberculosis which has a low base rate, it would be irrational to sa that a patient is likel to be infected, with high uncertaint. However, it would be rational to sa that the patient is probabl not infected, with high uncertaint low confidence. 20

21 Multinomial domain Generalisation of binar domain Set of eclusive and ehaustive singletons. Eample domain: ={ 1, 2, 3, 4 }, =

22 Multinomial Opinions Domain: ={ 1 k } Random variable Multinomial opinion: ω = b, u, a Belief mass distribution b where u + Σb = 1 b is belief mass on Uncertaint mass: u is a single value in range [0,1] Base rate distribution a where Σa = 1 a is base rate of Projected probabilit: P = b + a u 22

23 Opinion tetrahedron ternar domain u - ais Projector Opinion ω 2 ais 3 ais P Projected probabilit distribution a Base rate distribution 1 ais 23

24 Dirichlet PDF representation Dir p = Γ k i= 1 k i= 1 Γ α k i= 1 α i i p i α i 1 Σ p i = 1 α i = r i + W a i r i : # observations of i a i : base rate of i E : Epected proba. distr. Densit Dirp Trinomial Dirichlet Probabilit Densit Function E = P Eample: 6 red balls 1 ellow ball 1 black ball 24

25 25 Multinomial Opinion Dirichlet PDF Dirichlet PDF evidence parameters: r, a Multinomial opinion parameters: b, u, a Op Dir: Dir Op: = + = 1 b u u b W r + = + = r W W u r W r b W = 2

26 Non-informative prior weight: W Value normall set to W = 2. When W is equal to the frame cardinalit, then the prior Dirichlet PDF is a uniform. Normall required that the prior Beta is uniform, which dictates W = 2 Beta PDF is a binomial Dirichlet PDF Setting W > 2 would make Dirichlet PDF insensitive to new observations, which would be an inadequate model. 26

27 Prior trinomial Dirichlet PDF, W = 2 Eample: Urn with balls of 3 different colours. t1: Red t2: Yellow t3: Black Ternar a priori probabilit densit. 27

28 Posterior trinomial Dirichlet PDF posteriori probabilit densit after picking: 6 red balls t1 1 ellow ball t2 1 black ball t3 W = 2 28

29 Posterior trinomial Dirichlet PDF posteriori probabilit densit after picking: 20 red balls t1 20 ellow balls t2 20 black balls t3 W = 2 29

30 Posterior trinomial Dirichlet PDF posteriori probabilit densit after picking: 20 red balls t1 20 ellow balls t2 50 black balls t3 W = 2 30

31 Hper-Opinions Domain: ={ 1 k } P is the powerset of Hperdomain R = P \ {, } R is the reduced powerset of Hpervariable: R Hper opinion: ω = b, u, a Belief mass distribution: b b is belief mass on R Base rate distribution: a a is base rate of Proj. probabilit: P where where u = a u + a R j + b R a = 1 j b = 1 31 j

32 Hper Dirichlet PDF HDir p p 2 p 3 p 1 32

33 Opinions v. Fuzz membership functions Fuzz logic 250 cm Domain of fuzz categories Tall verage Short Fuzz membership functions 200 cm 150 cm 100 cm 50 cm Crisp measures 0 cm Subjective logic Domain of crisp categories Friendl aircraft Enem aircraft Civilian aircraft Subjective opinions ω Uncertain measures 33

34 Decision Making Under Vagueness and Uncertaint 34

35 Sharp and Vague Belief Mass Opinion ω M 1 : P 1 M 2 : P 2 M 3 : P 3 M 4 : M 5 : M 6 : R 1 2 P 4 P 5 P a 1 = 0.2 a 2 = 0.3 a 3 = 0.5 P Sharp belief mass Vague belief mass Focal uncertaint mass 35

36 36 Sharp belief mass: Vague belief mass: Focal uncertaint mass: dditivit: Mass-sum: = i i b b S = i i i b a b V u a u F = P F S S u b b + + =,, M F S S u b b =

37 Eample Vague Opinion Opinion ω b 6 = 0.8 a 1 = 0.33 u = 0.2 a 2 = 0.33 a 3 = 0.33 R HDir p p p p 1 V b 1 = 0 V b 2 = 0. 4 V 3 = 0. 4 b F u 1 = F u 2 = F 3 = u 37

38 Total Mass-Sum Total sharp belief mass: b TS = Χ b Total vague belief mass: Total uncertaint mass: b TV u = C b Χ dditivit: b TS TS + b + u = 1 M T : P Sharp belief mass Vague belief mass Focal uncertaint mass 38

39 Utilit-Normalised Mass-Sum Utilit distribution: Epected utilit: λ L = λ P Greatest utilit: + λ = ma[ λ ] Utilit-normalised probabilit: P N L = λ + Utilit-normalised sharpness: b NS S λ b = λ + U-N Mass-Sum: M N = NS NS NF b, b, u 39

40 Eample U-N Mass-Sum b 1 = 0.7, a 1 = 0.5 b ω Y 1 = 0.4, a Y 1 = 0.5 b 2 = 0.1, a 2 = 0.5 b Y 2 = 0.2, a Y 2 = 0.5 u = 0.2 λ 2 = 1000 ω Y u Y = 0.4 λ Y 2 = 500 Opt., Opt.Y, M 2 : M 2 : Y P 2 PY 2 Mass-sums Opt., Opt.Y, M N 2 : M N Y 2 : P N 2 Utilit-normalised mass-sums P YN P Sharp belief mass Focal uncertaint mass 40

41 Consider mass-sums for all decision options Decision Criteria Compare utilit-normalised probabilities Equal Compare belif sharpness Equal Compare focal uncertainties Different Different Different Select option with the greatest utilit-normalised probabilit Select option with the greatest belief sharpness Select option with the least focal uncertaint Equal Difficult decision 41

42 The Ellsberg Parado Traditional probabilit theor is unable to eplain tpical human decision-making behaviour. Probabilities do not epress degrees of vagueness and uncertaint. The Ellsberg parado involves vagueness. Probabilit theor can not eplain the results of the Ellsberg eperiment. Subjective opinions epress degrees of vagueness and uncertaint Subjective opinions eplain the results of the Ellsberg as rational. Parado solved. 42

43 Ellsberg Parado 90 coloured balls 1 : red balls 30 2 : black balls 3 : ellow balls 6 : black or ellow 60 R / /3 0 Game 1 Game 2 Option Option B Option λ 1 = 100 λ 1 = 0 λ 1 = 100 λ 2 = 0 λ 2 = 100 λ 2 = 0 λ 3 = 0 λ 3 = 0 λ 3 = 100 Option B λ 1 = 0 λ 2 = 100 λ 3 = 100 People choose Opt.1 People choose Opt.2.B 43

44 The Ellsberg Parado Eplained Game 1 Opt.1, M 1 : Opt.1B, : P 2 M 2 Game P Opt.2, M 5 : P 5 Opt.2B, M 6 : P P P Sharp belief mass Vague belief mass 44

45 Subjective Logic Operators 45

46 Homomorphic correspondence Generalisation Generalisation Binar logic Booleans Truth tables Homomorphic i.c.o. probabilit 0 or 1 Probabilistic logic Probabilities PL operators Homomorphic i.c.o. dogmatic multi-nomial opinions Subjective logic Opinions ω SL operators Homomorphic in case of absolute binomial opinions Pω ω = Pω Pω for probabilistic multiplication Bω ω = Bω Bω for Boolean conjunction 46

47 Subjective logic operators 1 Opinion operator name Opinion Logic Logic operator name operator operator smbol smbol ddition + UNION Subtraction - \ DIFFERENCE Complement NOT Projected probabilit P n.a. n.a. Multiplication ND Division / UN-ND Comultiplication Codivision п п OR UN-OR 47

48 Subjective logic operators 2 Opinion operator name Opinion Logic Logic operator name operator smbol operator smbol Transitive discounting : TRNSITIVITY Cumulative fusion n.a. veraging fusion n.a. Constraint fusion & n.a. Inversion, ~ ~ Baes theorem φ Conditional deduction CONTRPOSITION DEDUCTION Modus Ponens Conditional abduction ~ ~ BDUCTION Modus Tollens 48

49 ddition Notation ω 1 = ω 2 + ω 1 2 Probabilit version: p 1 2 =p 1 + p 2 Commutative and associative. Increases vagueness 49

50 Eample addition u u u PLUS = d b d b d ap P a P a b Opinion about 1 belief 0.20 disbelief 0.40 uncertaint 0.40 base rate 0.25 probabilit 0.30 Opinion about 2 belief 0.10 disbelief 0.50 uncertaint 0.40 base rate 0.50 probabilit 0.30 Opinion about 1 2 belief 0.30 disbelief 0.30 uncertaint 0.40 base rate 0.75 probabilit

51 Subtraction Notation: ω \ = ω = ω ω Probabilit version: p 1 2 \ 2 = p 1 =p 1 2 p 2 Reduces vagueness 51

52 Eample subtraction u u u MINUS = d b d b d a P a P P a b Opinion about 1 2 belief 0.70 disbelief 0.10 uncertaint 0.20 base rate 0.75 probabilit 0.85 Opinion about 2 belief 0.50 disbelief 0.30 uncertaint 0.20 base rate 0.25 probabilit 0.55 Opinion about 1 belief 0.20 disbelief 0.60 uncertaint 0.20 base rate 0.50 probabilit

53 Complement Notation: Corresponds to NOT in binar logic ω = ω Involutive: ω = ω Probabilit version: p=1 - p 53

54 Cartesian product of frames Multiplication assumes a Cartesian product. Product set has Cardinalit = Y. Coarsening needed as part of computation. Basis for product and coproduct. Y = Y,,,, product: coproduct: 54

55 Binomial multiplication conjunction ω = ω ω Notation: Probabilit version: p =p p Commutative and associative. Corresponds to ND and probabilit product. Y = Y,,,, 55

56 Eample multiplication u u u ND = d b d b d a P a P a P b Opinion about belief 0.75 disbelief 0.15 uncertaint 0.10 base rate 0.50 probabilit 0.80 Opinion about belief 0.10 disbelief 0.00 uncertaint 0.90 base rate 0.20 probabilit 0.28 Opinion about ND belief 0.15 disbelief 0.15 uncertaint 0.70 base rate 0.10 probabilit

57 Binomial comultiplication disjunction Notation: ω = ω Commutative and associative. Corresponds to OR and probabilit coproduct. Probabilit version: p = p + p pp ω п Y = Y,,,, 57

58 Eample comultiplication u u u OR = d b d b d a P a P a P b Opinion about belief 0.75 disbelief 0.15 uncertaint 0.10 base rate 0.50 probabilit 0.80 Opinion about belief 0.35 disbelief 0.00 uncertaint 0.65 base rate 0.20 probabilit 0.48 Opinion about OR belief 0.84 disbelief 0.06 uncertaint 0.10 base rate 0.60 probabilit

59 Non-distributivit of product and coproduct Binomial product is non-distributive on binomial coproduct: ω nor for probabilities: ω z z p z p z z z Onl holds for binar logic: z = z 59

60 Subjective Trust Networks 60

61 Trust transitivit lice Thanks to Bob s advice, lice trusts Eric to be a good mechanic. 4 Indirect functional trust Eric Direct referral trust Bob has proven to lice that he is knowledgeable in matters relating to car maintenance. 3 dvice 2 Bob 1 Direct functional trust Eric has proven to Bob that he is a good mechanic. 61

62 Functional trust derivation requirement lice Bob Claire Eric 2 adv. 1 referral trust 2 adv. 1 referral trust functional trust 3 1 functional trust Functional trust derivation through transitive paths requires that the last trust edge represents functional trust or an opinion and that all previous trust edges represent referral trust. Functional trust can be an opinion about a variable. 62

63 Trust transitivit characteristics Trust is diluted in a transitive chain. B C E adv. adv. trust trust trust diluted trust Computed with discounting/transitivit operator of SL Graph notation: [, E] = [; B] : [B; C] : [C, E] SL notation: ω ; B; C E = ω B ω B C ω C E 63

64 Trust Fusion Combination of serial and parallel trust paths Bob Trusting part lice David 1 func-trust Eric Trusted part Claire derived func-trust 3 Graph notation: SL not.: ω [, E] = [;B] : [B;D] [;C] : [C;D] : [D,E] [ ; B; D] [ ; C; D] E = ω B ω B D ω C ω C D ω D E 64

65 Discount and Fuse: Dilution and Confidence adv. trust trust adv. B C trust trust D confident diluted trust trust Discounting dilutes trust confidence Fusion strengthens trust confidence 65

66 Incorrect trust / belief derivation 3 B 1 2 D C 3 Beware! 1 2 incorrect belief 4 Perceived: [, B] : [B, ] [, C] : [C, ] ref. trust belief / func. trust advice Hidden: [, B] : [B, D] : [D, ] [, C] : [C, D] : [D, ] 66

67 Hidden and perceived topologies Perceived topolog: Hidden topolog: B B D C C D [, B] : [B, ] [, C] : [C, ] [, B] : [B, D] : [D, ] [, C] : [C, D] : [D, ] D, E is taken into account twice 67

68 Correct trust / belief derivation 1 1 B D C 1 correct belief 2 SFE ref. trust belief / func. trust advice Perceived and real topologies are equal: [; B] : [B; D] [; C] : [C; D] : [D, ] 68

69 Computing discounted trust ω B B B B ω ω ; B u u u t b B u B ω B d B P B ab t 2 B a B b B P B ω B u B d s trust in B B s opinion about s derived opinion about 1 = 2 B a : B b : P : B ω : B B u : B d : P B 1 69

70 Eample: Weighing testimonies Computing beliefs about statements in court. J is the judge. W 1, W 2, W 3 are witnesses providing testimonies. J W 1 W 2 W 3 statement ω J ; W1 J ; W2 J ; W3 70

71 Computational trust with subjective logic 71

72 Baesian Reasoning 72

73 py Joint Probabilities and Marginalisation Conditionals = py 1 py 2 py k p 1 1 p 1 2 p 1 k p 2 1 p 2 2 p 2 k p l 1 p l 2 p l k Joint probabilities... py... p p 1 p 2... p k p 1, 1 p 1, 2 p 1, k p 2, 1 p 2, 2 p 2, k... p l, 1 p l, 2 p l, k marg. p Evidence marg. py 73

74 Joint and Marginal Opinions Joint base rate distribution: a Y = P Y a Joint opinion: ω Y = ωy ω Marginal opinion on : Marginal opionion on Y: Deduced opinion on Y: ω = ω [ ] ω ω [ Y ] Y ω = ω ω Y Y 74

75 Deduction and bduction Parent node ω ω ~ Y Causal conditionals Child node Y ω Y Deduction ω Y bdduction 75

76 Deduction visualisation Evidence pramid is mapped inside hpothesis pramid as a function of the conditionals. Conclusion opinion is linearl mapped Opinions on parent ω u Opinions on child Y ω Y u Y ω Y ω Y 2 ω b 2 ω Y 3 ω Y 1 b 2 b 3 b 1 b 3 b 1 76

77 Deduction online operator demo 77

78 78 Baes Theorem Traditional statement of Baes theorem: Baes theorem with base rates: Marginal base rates: Baes theorem with marginal base rates p p p p = p a p a a + = a p a p = p a p a p a p + = p a p a p a p + =

79 79 The Subjective Baes Theorem Inversion of conditional opinions Y,,, ~ ~ a ω ω ω ω = φ ~ Y Binomial: Y, ~ Y Y a ω ω = φ ~ Y Multinomial:

80 Subjective Baes theorem Inversion Visualisation ntecedent 3 ω Y Consequent 3 ω ~ 3 u verte ω Conditionals ω Y Inversion of conditional opinions Conditionals 2 2 ω Y 1 ω Y u verte 2 ω ~ 2 ω ~ 1 1 ω ~ Y 3 ωy ω Y u Y verte Consequent Y u Y verte ntecedent Y 2 ω Y

81 Subjective Baes Theorem and Uncertaint u verte ω u verte ω -triangle Convergence conditionals Y-triangle ω ~ ω ω a ω ~ Figure shows effect of repeated conditional inversion with the subjective Baes theorem Uncertaint increases and converges to uncertaint-maimised conditional opinions a Initial conditionals 81

82 Deduction and abduction notation ω ω Y ω Y Y Deduction ω = ω ω Y Y ω Y ω ~ Y a ~ ~ ω Y Y Y, = ω ~ Y φ ωy, a = ω Y ω ~ Y ω = ω ω Y ω Y Y bdduction 82

83 83 Eample: Medical reasoning Medical test reliabilit determined b: true positive rate where : infected false positive rate : positive test Baes theorem: Probabilistic model hides uncertaint Use subjective Baes theorem to determine ω infected GP derives ω infected positive and ω infected negative Finall compute diagnosis ω infected test result Medical reasoning with SL reflects uncertaint ~ p a p a p a p p p p + = = ~ ~ p p, ~ Y Y a ω ω = φ ~

84 bduction Online operator demo 84

85 The General Idea of Subjective Networks Baesian Networks Subjective Logic Subjective Baesian Networks Subjective Trust Networks Subjective Networks 85

86 Eample SN Model B ω B ω B C C ω ω C ω Y ω Z Y ω Z Y Z ω B C Z = ωb ω ωc ω ωy ωz Y 86

87 Subjective Networks S ub j e ct i v e N et w or k Subjective Trust Network: Subjective Baesian Network: B Z C Y D Frame of agents Frame of variables Legend: Trust relationship Belief relationship Conditional relationship 87

88 New Book on Subjective Logic 88