Extreme probability distributions of random/fuzzy sets and p-boxes Alberto Bernardini; University of Padua, Italy Fulvio Tonon; University of Texas, USA

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

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

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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?. gt 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
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Imprecise Probability (Walley, 1991)
Given ELOWfi and ELOWfi, fi ? K What can
we say about probabilities of events in S?
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Objectives

• To give expressions for ?E in special cases
• Choquet capacities of order ?2
• Random sets
• Fuzzy sets
• P-boxes

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Choquet capacities of order ?2
Order 2
Set function ? P(S) ? ?0, 1? ? (?) 0, ?(S)
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Alternate Choquet Capacity of order k 2
?UPP(Ti) 1- ?(Tic)
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Choquet capacities of order ?2
Coherent upper and lower probabilities
Coherent upper and lower previsions
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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

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Choquet capacities of order ?2
Calculation of expectation for f S? YyL, yR ?
?
yR
yL
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Choquet capacities of order ?2
Calculation of expectation bounds for f S?
YyL, yR ? ? when ? is given
When f attains values ?1 yRgt ?2 gt gt ?n yL
Same as reordering S gt f becomes monotonically
decreasing
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Random sets
Family of n focal elements Ai ? S with weights
m(Ai) m(?)0 ?i m(Ai)1
? is the convex hull of EXT Expectations may be
calculated by reordering S
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Random sets, Bel Pla
Coherent upper and lower probabilities
Choquet capacities, k?2
Choquet capacities, k?
Coherent upper and lower previsions
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Fuzzy sets
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Probability (P-) boxes
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Probability (P-) boxes
All P ? ?E satisfy the constraints
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Probability (P-) boxes
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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 gt 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