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Extreme probability distributions of random/fuzzy sets and p-boxes Alberto Bernardini; University of Padua, Italy Fulvi

Extreme probability distributions of random/fuzzy sets and p-boxes Alberto Bernardini; University of Padua, Italy Fulvio Tonon; University of Texas, USA. Outline. Review of Imprecise Probability Objective Set of probability distributions for: Choquet capacities Random sets Fuzzy sets

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Extreme probability distributions of random/fuzzy sets and p-boxes Alberto Bernardini; University of Padua, Italy Fulvi

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  1. Extreme probability distributions of random/fuzzy sets and p-boxesAlberto Bernardini; University of Padua, ItalyFulvio Tonon; University of Texas, USA

  2. Outline • Review of Imprecise Probability • Objective • Set of probability distributions for: • Choquet capacities • Random sets • Fuzzy sets • P-boxes

  3. Imprecise Probability (Walley, 1991) Finite probability space (, F, P), F is the -algebra generated by a finite partition of  into elementary events (or singletons) S = s1, s2…, sj,… sn. => Probability space is fully specified by the probabilities P(sj) Consider bounded and point-valued functions (gambles) fi: S For a specific precise probability distribution P(sj), the prevision is equivalent to the linear expectation: T = 1 if sjT, 0 if sjT

  4. Can E be arbitrary? NO: 2 basic conditions: • E should be non-empty (avoid sure loss) • Given lower/upper bounds should be the same as obtained with E (coherence) Given ELOW[fi] and ELOW[fi],fiK; What can we say about probabilities of events in S? Imprecise Probability (Walley, 1991)

  5. Objectives • To give expressions for E in special cases: • Choquet capacities of order 2 • Random sets • Fuzzy sets • P-boxes

  6. Order k Order 2 Choquet capacities of order 2 Set function : P(S)  0, 1 :  () = 0, (S)= 1 Alternate ChoquetCapacity of order k = 2:UPP(Ti) = 1- (Tic)

  7. Choquet capacities, k2 Choquet capacities of order 2 Coherent upper and lower probabilities Coherent upper and lower previsions

  8. Choquet capacities of order 2 • Given (.): • Consider a permutation  of the elements of S = s1, s2…, sj,… sn • Construct probability distribution P: • Repeat 1) and 2) for all n! permutations • The set, EXT, of P so constructed is the set of extreme points of E • E is the convex hull of EXT

  9. Choquet capacities of order 2 Calculation of expectation for f: S Y=[yL, yR]   yR yL

  10. Choquet capacities of order 2 Calculation of expectation bounds for f: S Y=[yL, yR]   when  is given When f attains values: 1= yR> 2 >… > n = yL : Same as reordering S => fbecomes monotonically decreasing

  11. Random sets Family ofnfocal elementsAi  Swith weightsm(Ai) : m()=0; i m(Ai)=1 is the convex hull of EXT Expectations may be calculated by reordering S

  12. Bel, Pla Random sets, Bel & Pla Coherent upper and lower probabilities Choquet capacities, k2 Choquet capacities, k= Coherent upper and lower previsions

  13. m1 m1 m2 m3 A1 A2 A3 Fuzzy sets

  14. A1 A2 A3 A4 A5 Probability (P-) boxes

  15. Probability (P-) boxes All P  E satisfy the constraints

  16. Probability (P-) boxes

  17. Conclusions • The set of probability distributions compatible with a random set is equal to the convex hull of the extreme distributions obtained by permuting elements of S • Exact bounds on expectation of f may be calculated • By reordering the set S => f is monotonically decreasing • By using two elements in the set of extreme distributions • Fuzzy sets and p-boxes are particular indexable random sets • Random sets can be easily derived • Extreme distributions can be easily calculated

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