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Dempster-Shafer Theory SIU CS 537 4/12/11 and 4/14/11 Chet Langin Dempster, A. P. (1967). "Upper and Lower Probabilities Induced by a Multivalued Mapping.“ The Annals of Mathematical Statistics 38(2): 325-339. Shafer, G. (1976). A Mathematical Theory of Evidence, Princeton University Press. What is Dempster-Shafer? Dempster-Shafer (D-S or DS) Mathematical theory of evidence. Data fusion. Degree of belief. Generalization of Bayes theory. Sets. Mass, not probability. “Bel Function” – Belief function The D-S Environment • • • • Θ (Theta): Θ = {1 , 2 , 3 , … , } The elements are all mutually exclusive. All of the possible elements in the universe are in the set and so the set is exhaustive. Each subset of Θ can be interpreted as a possible answer to a question. Since the elements are mutually exclusive and the environment is exhaustive, there can be only one correct answer to a question. D-S Environment, Cont. Θ (Theta): Θ = {1 , 2 , 3 , … , } • All the possible subset of = {, , } → Fig. 5.6, Page 281 (Airliner, Bomber, Fighter). • An environment is called a Frame of Discernment where its elements may be interpreted as possible answers, and only one answer is correct. D-S Environment, Cont. Θ (Theta): Θ = {, , } • A set of size N has exactly 2 subsets, including itself, and these subsets define the Power Set ((Θ)): (Θ) = {∅, , , , , , , , , , , , } • The Power Set of the environment has as its elements all answers to the possible questions of the Frame of Discernment. D-S vs. Probability • In D-S Theory, the Degree of Belief in evidence is analogous to the mass of a physical object (mass of evidence supports a belief). Evidence measure ≡ amount of the mass ≡ ≡ Basic Probability Assignment (BPA). • Fundamental difference between D-S Theory and probability theory is the treatment of ignorance. = 1 ↔ Principle of indifference + ′ = 1 Non-belief vs. Ignorance • D-S does not force belief to be assigned to ignorance. Instead, the mass is assigned only to those subsets of the environment to which you wish to assign belief. • Not assigned belief ≡ no belief or non-belief. Should be associated with the environment Θ. Disbelief ≢ non-belief. 1 , = 0.7. 1 Θ = 0.3. • Every set in the Power Set of the environment which has a mass greater than zero is a Focal Element. D-S Mass • Mass is a function that maps each element of the Power Set into a real number in the [0,1] interval. : Θ → [0,1] By conversion: ∅ = 0 =1 ∀,∈(Θ) Combining Evidence First radar data: 1 , = 0.7 Second radar data: 2 = 0.9 1 Θ = 0.3 2 Θ = 0.1 D-S Rule of Combination 1 ⨁2 = 1 ()2 () ∩= Extends over all elements whose intersection ∩ = . ⨁ denotes the orthogonal sum or direct sum which is defined by summing the mass product intersections on the right-hand side of the rule. The new mass is a consensus because it tends to favor agreement rather than disagreement. Example Combination 2 1 , = 0.7 1 Θ = 0.3 3 ⟶ = 0.9 2 Θ = 0.1 0.63 , 0.07 0.27 Θ 0.03 Bomber = 0.63 + 0.27 = 0.90 Bomber or Fighter = 0.07 Non-belief = 0.03 Range of Belief 3 ({}) represents the belief of a bomber, only, but 3 ({, }) and 3 (Θ) imply additional information since their sets include a bomber ⟹ It is plausible that their orthogonal sums may contribute to a belief in the bomber. 0.9 + 0.07 + 0.03 = 1.0 ⟹ Plausible that it might be a bomber. What Would Make Plausibility < 1? 3 = 0.1 (Airliner) Evidential Interval The true Range of Belief is somewhere in the range of 0.9 to 1.0. Also called the Evidential Interval. The lower bound (0.9) is called Support (spt) in evidential reasoning. It is called Bel in DS Theory. The upper bound (1.0) is called Plausibility (pls). In general: 0 ≤ Bel ≤ Pls ≤ 1 (See Table 5.5 in text) Example Evidential Intervals [1, 1] [0, 0] [0, 1] [Bel, 1] [0, Pls] [Bel, Pls] Completely True Completely False Completely Ignorant Tends to support Tends to refute Tends to both support & refute Bel vs. Bel() • Bel is belief, a part of the evidence interval. It refers to one set. • Bel() is a function that is the total belief of a set and all its subsets. Bel function ≡ belief measure Bel() Example Bel = () ⊆ Bel1 , = 1 , + 1 + 1 ({}) 0.7 + 0 + 0 = 0.7 All the mass that supports a set. Is more global. Combination of 2 Bel() The combination of 2 belief functions (as in mass) can be expressed in terms of orthogonal sums of the masses of a set and all its subsets: 1 ⨁2 1 ⨁2 , = 1 ⨁2 + 1 ⨁2 ∅ = 0.90 + 0 = 0.90 = 1 ⨁2 , + 1 ⨁2 = 0.07 + 0 + 0.90 = 0.97 + 1 ⨁2 The Normalization of Belief Suppose a third sensor is provided: 3 = 0.95 3 = 0.05 A New Table 3 = 0.95 3 Θ = 0.05 ⨁ ({}) . ⨁ ({, }) . ⨁ () . ∅0 ∅0 0.0285 0.045 , 0.0035 Θ 0.0015 1 ⨁2 ⨁3 1 ⨁2 ⨁3 1 ⨁2 ⨁3 , 1 ⨁2 ⨁3 Θ 1 ⨁2 ⨁3 ∅ = 0.0285 = 0.0450 = 0.0035 = 0.0015 = 0.0000 = 0.0785 ≠ 1 ‼! Normalization Divide each element by 1-k where k is defined for any set X and Y as: = 1 ()2 () ∩=∅ = 0.95 × 0.90 + 0.95 × 0.07 = 0.8550 + 0.0665 = 0.9215 Normalization, Cont. 1 − = 1 − 0.9215 = 0.0785 0.0285 1 ⨁2 ⨁3 = = 0.363 0.0785 0.0450 1 ⨁2 ⨁3 = = 0.573 0.0785 0.0035 1 ⨁2 ⨁3 , = = 0.045 0.0785 0.0015 1 ⨁2 ⨁3 = = 0.019 0.0785 0.0000 1 ⨁2 ⨁3 ∅ = = 0.000 0.0785 = 1.000 OK! New Evidential Interval Belief | Plausibility Belief | Plausibility | Disbelief