Outline. On Premise Evaluation On Conclusion Entailment. 1 Imperfection : Why and What. 2 Imperfection : How. 3 Conclusions

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2 Outline 1 Imperfection : Why and What 2 Imperfection : How On Premise Evaluation On Conclusion Entailment 3 Conclusions

3 Outline 1 Imperfection : Why and What 2 Imperfection : How On Premise Evaluation On Conclusion Entailment 3 Conclusions

4 Basic Inference P(x),P(X ) C(Y ) C(y) Classic Modus Ponens Premise and Implication entail Consequence

5 Basic Inference P(x),P(X ) C(Y ) C(y) Classic Modus Ponens Premise and Implication entail Consequence Is this model really applicable in practice?

6 Once upon a time... A well known story... [Bezdek] : thirsty(x ) has(x, Bottle) lethal(bottle) drink(bottle)

7 Once upon a time... A well known story... [Bezdek] : thirsty(x ) has(x, Bottle) lethal(bottle) drink(bottle) Lack of information Ill-defined information Erroneous information...

8 What is Imperfection? Imperfection Imperfection, be it Imprecision or Uncertainty, pervades... systems that attempt to provide an accurate model of the real world P.Smets, 1999

9 What is Imperfection? Imperfection Imperfection, be it Imprecision or Uncertainty, pervades... systems that attempt to provide an accurate model of the real world P.Smets, 1999 Uncertainty Uncertainty is a condition where Boolean truth values are unknown, unknowable, or inapplicable... W3C Incubator Group on Uncertainty Reasoning for the Web, 2005

10 What is Imperfection? Imperfection - a negative definition Uncertainty/Imperfection is the opposite of preciseness and certainty, i.e. of what Boolean logic models

11 Using Imperfection Rules should handle imperfection, not ignore it Benefits Conciseness Robustness Drawbacks Complexity Correctness and Coherence

12 More remarks My three cents on predictive modelling technologies...

13 More remarks My three cents on predictive modelling technologies... Knowledge Business Rules Predictive Models

14 More remarks Knowledge Business Rules Predictive Models

15 More remarks Knowledge Hard AI Soft AI

16 More remarks Knowledge Perfect AI Imperfect AI

17 More remarks Knowledge Perfect AI Imperfect AI We have a (serious) problem...

18 More remarks Knowledge Perfect AI Imperfect AI We have a (serious) problem... Business Rules Rules

19 More remarks Knowledge Perfect AI Imperfect AI We have a (serious) problem... Business Rules Rule-Based Programming

20 Outline 1 Imperfection : Why and What 2 Imperfection : How On Premise Evaluation On Conclusion Entailment 3 Conclusions

21 An Ontology for Imperfection Uncertainty Nature Derivation Type Model Aleatory Episthemic Subjective Objective Inconsistency Vagueness Incompleteness FuzzySets Ambiguity Randomness Probability Belief RandomSets RoughSets more...

22 An Ontology for Imperfection Uncertainty Uncertainty / Confidence Factors Nature Derivation Type Model Aleatory Episthemic Subjective Objective Inconsistency Vagueness Incompleteness FuzzySets Ambiguity Randomness Probability Belief RandomSets RoughSets more...

23 An Ontology for Imperfection Uncertainty Uncertainty / Frequentist Probability Nature Derivation Type Model Aleatory Episthemic Subjective Objective Inconsistency Vagueness Incompleteness FuzzySets Ambiguity Randomness Probability Belief RandomSets RoughSets more...

24 An Ontology for Imperfection Uncertainty Uncertainty / Bayesian Probability Nature Derivation Type Model Aleatory Episthemic Subjective Objective Inconsistency Vagueness Incompleteness FuzzySets Ambiguity Randomness Probability Belief RandomSets RoughSets more...

25 An Ontology for Imperfection Uncertainty Vagueness / Fuzzy Logic Nature Derivation Type Model Aleatory Episthemic Subjective Objective Inconsistency Vagueness Incompleteness FuzzySets Ambiguity Randomness Probability Belief RandomSets RoughSets more...

26 Generalized Inference P(x),P(X ) C(Y ) C(y) Classic Modus Ponens Premise and Implication entail Consequence

27 Generalized Inference Φ(...,A j (x)/ε j,... ),P(X ) C(Y ) C(y) Premise Atomic constraints are evaluated General, pluggable Evaluators A Degree is returned

28 Generalized Inference Φ(...,A j (x)/ε j,... )/ε P,P(X ) C(Y ) C(y) Premise Atomic constraints are evaluated General, pluggable Evaluators A Degree is returned Premise Atoms are aggregated in formulas using generalized logic Connectives evaluated by Operators

29 Generalized Inference Implication Implication has a Degree often given a priori P(x)/ε P, (X,Y ) /ε C(y)

30 Generalized Inference P(x)/ε P, (X,Y ) /ε C(y)/ε C Implication Implication has a Degree often given a priori Modus Ponens MP computes the Degree of the Consequence

31 Generalized Inference P 1, 1 C 1 /ε C1,..., P n, n Cn/ε C n C(y)/ε C Merging multiple sources Multiple premises for the same conclusion Solve conflicts Handle missing values

32 On Premise Evaluation Outline 1 Imperfection : Why and What 2 Imperfection : How On Premise Evaluation On Conclusion Entailment 3 Conclusions

33 On Premise Evaluation Generalized Inference - Imperfect Facts Φ(...,A j (x)/ε j,... ),P(X ) C(Y ) C(y) Premise : (Complex) Formulas Atomic constraints are evaluated A Degree is returned

34 On Premise Evaluation Generalized Degrees Degrees generalize the boolean true/false truth: compatibility with a prototype probability: ratio of relevant events over total belief: opinion in assuming a property to be true. possibility: disposition towards accepting a situation to be true. confidence: strength of an agent s belief in a statement. Different models, including: τ Simple ε Interval ϕ ε Type-II degrees

35 On Premise Evaluation Uncertainty : Frequentist Probability p(x 1 ) p(x 2 ) p(x3 ) p(x 4 ) X Objective Probabilities Repeated trials Random Variables 1 out of N Expected value Applications Predict unobservable events What-if scenarios Drawbacks High amount of experimental data

36 On Premise Evaluation Uncertainty : Frequentist Probability p(x 1 ) p(x 2 ) p(x3 ) p(x 4 ) X Objective Probabilities Repeated trials Random Variables 1 out of N Expected value Applications Predict unobservable events What-if scenarios Drawbacks High amount of experimental data Evaluation: somepredicate( X /y ) p(x = y)

37 On Premise Evaluation Uncertainty : Bayesian Probability X Subjective Probabilities Non-repeatable events Prior, Conditional and Posterior Parametric Models Gaussian Poisson Beta, Gamma... Applications Express belief on the outcome of an event Drawbacks Model selection Parameter estimation

38 On Premise Evaluation Uncertainty : Bayesian Probability X Subjective Probabilities Non-repeatable events Prior, Conditional and Posterior Parametric Models Gaussian Poisson Beta, Gamma... Applications Express belief on the outcome of an event Drawbacks Model selection Parameter estimation How to build that belief? Evidence: p(a b) p(a) p(b a) p(b)

39 On Premise Evaluation Vagueness : Many-Valued and Fuzzy Logic µ(x) very low low medium high very high Partial membership Domains Tertium datur Fuzzy sets define predicates Quantitative domain Qualitative property Applications Gradual Properties Drawbacks Don t overlook definitions

40 On Premise Evaluation Vagueness : Many-Valued and Fuzzy Logic µ(x) very low low medium high very high Partial membership Domains Tertium datur Fuzzy sets define predicates Quantitative domain Qualitative property Applications Gradual Properties Drawbacks Don t overlook definitions Approximate relations/functions

41 On Premise Evaluation Vague Uncertainty : Possibility µ(x) very low low medium high very high Possibility Not probability! Linguistic Variables Vague statements Fuzzy sets as values Applications Manage vague facts Drawbacks Defuzzification

42 On Premise Evaluation Examples I?- age(davide,x) {X /20 : 30%, X /30 : 60%}

43 On Premise Evaluation Examples I?- age(davide,x) {X /20 : 30%, X /30 : 60%}?- age(davide,x) p(x ) N (30, 2)

44 On Premise Evaluation Examples I?- age(davide,x) {X /20 : 30%, X /30 : 60%}?- age(davide,x) p(x ) N (30, 2) age(davide,30) {old(davide)/0.25, young(davide)/0.75}

45 On Premise Evaluation Examples I?- age(davide,x) {X /20 : 30%, X /30 : 60%}?- age(davide,x) p(x ) N (30, 2) age(davide,30) {old(davide)/0.25, young(davide)/0.75} young(davide).

46 On Premise Evaluation Examples II?- hasdisease(davide,x) {X /cold : 30%, X /allergy : 10%, X /itchynose : 60%}

47 On Premise Evaluation Examples II?- hasdisease(davide,x) {X /cold : 30%, X /allergy : 10%, X /itchynose : 60%} hasdisease(davide,cold) {true/60%, false/20%} hasdisease(davide,allergy) {true/30%, false/70%} hasdisease(davide,itchy nose) {true/40%}

48 On Premise Evaluation Generalized Inference Φ(...,A j (x)/ε j,... )/ε P,P(X ) C(Y ) C(y) Premise Atomic constraints are evaluated General, pluggable Evaluators A Degree is returned Premise Atoms are aggregated in formulas using generalized logic Connectives evaluated by Operators

49 On Premise Evaluation Custom Operators {(Args), Degree} n Degree Define connectives (,,,... ) Return a Degree

50 On Premise Evaluation Custom Operators {(Args), Degree} n Degree Define connectives (,,,... ) Return a Degree Truth-functionality?

51 On Premise Evaluation Custom Operators {(Args), Degree} n Degree Define connectives (,,,... ) Return a Degree Truth-functionality? Vagueness Truth-functional Three main families (...and many others) Uncertainty Not Truth-functional... (... unless assumptions are made)

52 On Premise Evaluation Examples III?- hasdisease(davide,x) {X /cold : 30%, X /allergy : 10%, X /itchynose : 60%} hasdisease(davide, cold) hasdisease(davide, allergy)

53 On Premise Evaluation Examples III?- hasdisease(davide,x) {X /cold : 30%, X /allergy : 10%, X /itchynose : 60%} hasdisease(davide, cold) hasdisease(davide, allergy) Mutual exclusion : p(cold)+p(allergy) = 40%

54 On Premise Evaluation Examples IV hasdisease(davide,cold) {true/60%, false/20%} hasdisease(davide,allergy) {true/30%, false/70%} hasdisease(davide, cold) hasdisease(davide, allergy)

55 On Premise Evaluation Examples IV hasdisease(davide,cold) {true/60%, false/20%} hasdisease(davide,allergy) {true/30%, false/70%} hasdisease(davide, cold) hasdisease(davide, allergy) Independence? : p(cold) p(allergy) = ( )%

56 On Premise Evaluation Examples IV hasdisease(davide,cold) {true/60%, false/20%} hasdisease(davide,allergy) {true/30%, false/70%} hasdisease(davide, cold) hasdisease(davide, allergy) Independence? : p(cold) p(allergy) = ( )% Non-interactivity? : p(cold) p(allergy) = max{60, 30}%

57 On Premise Evaluation Examples IV hasdisease(davide,cold) {true/60%, false/20%} hasdisease(davide,allergy) {true/30%, false/70%} hasdisease(davide, cold) hasdisease(davide, allergy) Independence? : p(cold) p(allergy) = ( )% Non-interactivity? : p(cold) p(allergy) = max{60, 30}% Non-Independence? : we need e.g. p(cold allergy)

58 On Conclusion Entailment Outline 1 Imperfection : Why and What 2 Imperfection : How On Premise Evaluation On Conclusion Entailment 3 Conclusions

59 On Conclusion Entailment Generalized Inference P(x)/ε P, (X,Y ) /ε C(y)/ε C Implication Implication has a Degree often given a priori Modus Ponens MP computes the Degree of the Consequence

60 On Conclusion Entailment Implication and Deduction Two operators A B vs A B

61 On Conclusion Entailment Implication and Deduction Two operators A B vs A B Vagueness Gradual implications Gradual rules... Uncertainty Bayesian inference Distribution Semantics...

62 On Conclusion Entailment Examples V hasdisease(x, cold) sneeze(x ) Vague: the more serious the cold, the stronger the sneeze µ(hasdisease(x, cold)) µ(hasdisease(x, cold) sneeze(x )) Statistical : given that you have cold, you ll probably sneeze p(hasdisease(x, cold)) p(sneeze(x ) hasdisease(x, cold))/p(sneeze(x )) Epistemical : is it true that cold causes sneeze? p(hasdisease(x, cold)) p(sneeze(x )

63 On Conclusion Entailment Generalized Inference P 1, 1 C 1 /ε C1,..., P n, n Cn/ε C n C(y)/ε C Merging multiple sources Multiple premises for the same conclusion Solve conflicts Handle missing values

64 On Conclusion Entailment Generalized Inference P 1, 1 C 1 /ε C1,..., P n, n Cn/ε C n C(y)/ε C Merging multiple sources Multiple premises for the same conclusion Solve conflicts Handle missing values Usually or-like

65 On Conclusion Entailment Uncertainty : Dirichlet model Dirichlet N (p β) = N 1 Γ j=0 N 1 β j + N Γ (β j + 1) j=0 N 1 j=0 p β j j Subjective Discrete Probabilities Confidence Belief from experience Adds variance to probabilities Applications Incremental, adaptive systems Drawbacks Bootstrap

66 On Conclusion Entailment Uncertainty : Dirichlet model Dirichlet N (p β) = N 1 Γ j=0 N 1 β j + N Γ (β j + 1) j=0 N 1 j=0 p β j j Subjective Discrete Probabilities Confidence Belief from experience Adds variance to probabilities Applications Incremental, adaptive systems Drawbacks Bootstrap Consider source reliability when merging

67 On Conclusion Entailment Dempster-Shafer Theory and TBM m(a) = 1 1 K B C=A m 0 (B) m 1 (C) Uncertain Probabilities Belief vs Plausibility Set-valued events Ignorance Inconsistency Applications Incremental, evidence-driven systems Drawbacks Complexity

68 Outline 1 Imperfection : Why and What 2 Imperfection : How On Premise Evaluation On Conclusion Entailment 3 Conclusions

69 Conclusions Rules should handle imperfection, not ignore it

70 Conclusions Rules should handle imperfection, not ignore it Expressiveness Declarativeness Integration BUT

71 Conclusions Rules should handle imperfection, not ignore it Expressiveness Declarativeness Integration BUT Imperfection must be modelled correctly