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OUTLINES

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. OUTLINES. INTRODUTION BELIEF FRAMES OF DISCERNMENT COMBINIG THE EVIDENCE ADVANTAGES OF DS THEORY DISADVANTAGES OF DS THEORY

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OUTLINES

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  1. IN THE NAME OF ALLAHDECISION MAKING BY USINGTHE THEORY OF EVIDENCESTUDENTS:HOSSEIN SHIRZADEH, AHAD OLLAH EZZATISUPERVISOR:Prof. BAGERI SHOURAKISPRING 2009

  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

  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,…

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

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

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

  7. INTRODUCTIONTHE 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

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

  9. INTRODUCTIONTHE IDEA OF CHANCE • 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

  10. INTRODUCTIONCHANCES 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

  11. INTRODUCTIONCHANCES 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

  12. INTRODUCTIONBAYESIAN 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

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

  14. INTRODUCTIONBAYESIAN 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

  15. INTRODUCTIONBAYESIAN THEORY OF PARTIAL BELIEFThe Representation of Ignorance In Evidence Theory • Belief functions • Little evidence: • Both the proposition and it’s negation have very low degrees of belief • Vacuous belief function

  16. INTRODUCTIONBAYESIAN THEORY OF PARTIAL BELIEFCombination vs. Conditioning • 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

  17. INTRODUCTIONBAYESIAN THEORY OF PARTIAL BELIEFThe Representation of Ignorance • 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

  18. Belief • 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 = {θ1, θ2, θ3}, evidence against {θ1} is considered to be evidence for {θ2, θ3}, and belief will be allotted accordingly)

  19. Frames of Discernment • Dempster - Shafer theory assumes a fixed, exhaustive set of mutually exclusive events • Θ = {θ1, θ2, ..., θn} • Same assumption as probability theory • Dempster - Shafer theory is concerned with the set of all subsets of Θ, known as the Frame of Discernment • 2Θ = {f, {θ1}, …, {θn}, {θ1,θ2}, …, {θ1, θ2, ... θn}} • Universe of mutually exclusive hypothesis

  20. Frames of Discernment • A subset {θ1, θ2, θ3} implicitly represents the proposition that one of θ1, θ2 or θn 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

  21. Combining the Evidence • 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.

  22. Advantages 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

  23. Disadvantages • 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.

  24. Basic Probability Assignment • The basic probability assignment (BPA), represented as m, assigns a belief number [0,1] to every member of 2Θsuch 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)

  25. Basic Probability AssignmentExample • 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. Belief Functions • Obtaining the measure of the total belief committed to A: • Belief functions can be characterized without reference to basic probability assignments:

  27. Belief Functions • For Θ = {A,B} • BPA is unique and can recovered from the belief function

  28. Belief Functions • Focal element • A subset is a focal element if m(A)>0 • Core • The union of all the focal elements. • Theorem

  29. Belief FunctionsBelief Intervals • Ignorance in DS Theory: • The width of the belief interval: • The sum of the belief committed to elements that intersectA, but are not subsets of A • The width of the interval therefore represents the amount of uncertainty in A, given the evidence

  30. Belief FunctionsDegrees of Doubt and Upper Probabilities • 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. 30

  31. Belief FunctionsDegrees of Doubt and Upper ProbabilitiesExample m({1, 2}) = – Bel({1}) – Bel({2}) + Bel({1, 2}) = – 0.1 – 0.2 + 0.4

  32. Belief Functions Bayesian Belief Functions • 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.

  33. Belief Functions Bayesian Belief FunctionsExample • Suppose

  34. DempsterRule of Combination • 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

  35. Dempster Rule of CombinationCombining Two Belief Functions • m1: basic probability assignment for Bel1 • A1,A2,…Ak: Bel1’s focal elements • m2: basic probability assignment for Bel2 • B1,B2,…Bl: Bel2’s focal elements

  36. Dempster Rule of CombinationCombining Two Belief Functions Probability mass measure of m1(Ai)m2(Bj) committed to

  37. Dempster Rule of CombinationCombining two Belief functions • The intersection of two strips m1(Ai) and m2(BJ) has measure m1(Ai)m2(BJ) , since it is committed to both Ai and to BJ , we say that the joint effect of Bel1 and Bel2 is to commit exactly to • The total probability mass exactly committed to A:

  38. Dempster Rule of CombinationCombining two Belief functionsExample

  39. Dempster Rule of Combination Combining two Belief functions • The only Difficulty • some of the squares may be committed to empty set • If Ai and Bjare focal elements of Bel1 and Bel2 and if then • The only Remedy: • Discard all the rectangles committed to empty set • Inflate the remaining rectangles by multiplying them with

  40. Dempster Rule of Combination The Weight of Conflict • 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 Bel1 and Bel2 commit probability to disjoint subsets Ai and Bj

  41. Dempster Rule of Combination The Weight of Conflict (cont.) • Bel1 , Bel2 not conflict at all: • k = 0, Con(Bel1, Bel2)= 0 • Bel1 , Bel2 flatly contradict each other: • does not exist • k = 1, Con(Bel1, Bel2) = ∞ • In previous example k = 0.15

  42. Dempster’s rule of combination • Suppose m1 and m2 are basic probability functions over Θ. Then m1⊕m2 is given by • In previous example

  43. Dempster Rule of Combination An Application of DS Theory • Frame of Discernment: A set of mutually exclusive alternatives: • All subsets of FoD form:

  44. Dempster Rule of Combination An Application of DS Theory • Exercise deploys two “evidences in features” m1 and m2 • m1 is based on MEAN features from Sensor1 • m1 provides evidences for {SIT} and {¬SIT} ({¬SIT} = {STAND, WALK}) • m2 is based on VARIANCE features from Sensor1 • m2 provides evidences for {WALK} and {¬WALK } ({¬WALK } = {SIT, STAND})

  45. Dempster Rule of Combination An Application of DS Theory

  46. Dempster Rule of Combination An Application of DS TheoryCalculation of evidence m1 Bel(SIT) = 0.2 Pls(SIT) = 1 - Bel(¬SIT) = 0.5 Evidence z1=mean(S1) Concrete value z1(t) m1Concrete Value(SIT, ¬SIT, ) = (0.2, 0.5, 0.3)

  47. Dempster Rule of Combination An Application of DS TheoryCalculation of evidence m2 Evidence Bel(WALK) = 0.4 Pls(WALK) = 1-Bel(¬WALK) = 0.5 z2=variances(S1) Concrete value z2(t) m2Concrete Value(WALK, ¬WALK, ) = (0.4, 0.5, 0.1)

  48. Dempster Rule of Combination An Application of DS TheoryDS Theory Combination • Applying Dempster´s Combination Rule: • Due to m({})  Normalization with 0.92 (=1-0.08)

  49. Dempster Rule of Combination An Application of DS TheoryNormalized Values Belief(STAND) = 0.272 Plausibility(STAND) = 1 - (0.108+0.022+0.217+0.13) = 0.523

  50. Dempster Rule of Combination An Application of DS Theory Belief and Plausibility (SIT) Ground Truth: 1: Sitting; 2: Standing; 3: Walking

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