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Completely monotone outer approximations of lower probabilities on finite possibility spaces Erik Quaeghebeur SYSTeMS Research Group Ghent University Belgium { a , b , c , d } { a , b , c } { a , b , d } { a , c , d } { b , c , d } { a , b } {

  1. Completely monotone outer approximations of lower probabilities on finite possibility spaces Erik Quaeghebeur SYSTeMS Research Group Ghent University Belgium

  2. { a , b , c , d } { a , b , c } { a , b , d } { a , c , d } { b , c , d } { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } { a } { a } { b } { b } { c } { c } { d } { d } 0 /

  3. { a , b , c , d } P 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 5 5 1 1 8 8 2 2 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 3 3 3 1 1 2 8 8 8 4 4 { a } { a } { b } { b } { c } { c } { d } { d } 1 1 1 1 8 8 8 8 0 / 0

  4. { a , b , c , d } P = 1 2 · P + 1 2 · R 1 1 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 5 3 1 5 3 1 1 3 1 1 3 1 8 4 2 8 4 2 2 4 4 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 3 1 1 3 1 1 3 1 1 1 1 1 1 2 0 2 0 2 2 2 8 2 4 8 2 4 8 2 4 4 4 { a } { a } { b } { b } { c } { c } { d } { d } 1 1 1 1 1 1 1 1 4 0 4 0 4 0 4 0 8 8 8 8 0 / 0 0 0

  5. { a , b , c , d } P = 1 2 · P + 1 E P = 1 2 · E p + 1 2 · R 2 · E R 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 1 1 1 1 2 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 0 0 2 4 4 4 { a } { a } { b } { b } { c } { c } { d } { d } 1 1 1 1 4 4 4 4 linear–imprecise decomposition

  6. { a , b , c , d } P 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 1 1 1 1 2 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 0 0 2 4 4 4 { a } { b } { c } { d }

  7. { a , b , c , d } P 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 1 1 1 1 2 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 0 0 2 4 4 4 { a } { b } { c } { d } Recursive µ A = PA − ∑ B ⊂ A µ B M¨ obius transform

  8. { a , b , c , d } µ P 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 1 1 1 1 2 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 0 0 2 2 4 4 4 { a } { b } { c } { d } Recursive µ A = PA − ∑ B ⊂ A µ B M¨ obius transform

  9. { a , b , c , d } µ P 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 1 1 1 1 2 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 1 1 1 0 0 0 0 2 2 4 4 4 4 4 4 { a } { b } { c } { d } Recursive µ A = PA − ∑ B ⊂ A µ B M¨ obius transform

  10. { a , b , c , d } µ P 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 1 1 1 1 2 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 1 1 1 0 0 0 0 2 2 4 4 4 4 4 4 { a } { b } { c } { d } Recursive µ A = PA − ∑ B ⊂ A µ B M¨ obius transform

  11. { a , b , c , d } µ P 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } − 1 1 1 1 1 2 2 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 1 1 1 0 0 0 0 2 2 4 4 4 4 4 4 { a } { b } { c } { d } Recursive µ A = PA − ∑ B ⊂ A µ B M¨ obius transform

  12. { a , b , c , d } µ P 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } − 1 1 − 1 1 − 1 1 1 0 2 2 4 2 4 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 1 1 1 0 0 0 0 2 2 4 4 4 4 4 4 { a } { b } { c } { d } Recursive µ A = PA − ∑ B ⊂ A µ B M¨ obius transform

  13. { a , b , c , d } µ P 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } − 1 1 − 1 1 − 1 1 1 0 2 2 4 2 4 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 1 1 1 0 0 0 0 2 2 4 4 4 4 4 4 { a } { b } { c } { d } Recursive µ A = PA − ∑ B ⊂ A µ B M¨ obius transform

  14. { a , b , c , d } µ P 3 1 4 { a , b , c } { a , b , d } { a , c , d } { b , c , d } − 1 1 − 1 1 − 1 1 1 0 2 2 4 2 4 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 1 1 1 0 0 0 0 2 2 4 4 4 4 4 4 { a } { b } { c } { d } Recursive µ A = PA − ∑ B ⊂ A µ B M¨ obius transform

  15. { a , b , c , d } µ P 3 1 4 { a , b , c } { a , b , d } { a , c , d } { b , c , d } − 1 1 − 1 1 − 1 1 1 0 2 2 4 2 4 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 1 1 1 0 0 0 0 2 2 4 4 4 4 4 4 { a } { b } { c } { d } Recursive µ A = PA − ∑ B ⊂ A µ B M¨ obius transform PA = ∑ B ⊆ A µ B M¨ obius inverse

  16. { a , b , c , d } P 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 1 1 1 1 2 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 0 0 2 4 4 4 { a } { b } { c } { d } Iterative Rescaling Method

  17. { a , b , c , d } P ν 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } − 1 1 1 1 1 2 2 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 1 1 1 0 0 0 0 2 2 4 4 4 4 4 4 { a } { b } { c } { d } Recursive Iterative Rescaling Method M¨ obius transform

  18. { a , b , c , d } P ν 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } − 1 2 + 1 1 1 1 1 2 2 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 2 − 1 1 1 4 − 1 1 1 1 1 4 − 1 1 0 0 0 0 4 2 8 4 4 4 8 4 { a } { b } { c } { d } Rescale Iterative Rescaling Method when negative

  19. { a , b , c , d } P ν 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 1 1 1 1 0 2 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 1 1 1 0 0 0 0 4 2 8 4 4 4 8 4 { a } { b } { c } { d } Iterative Rescaling Method

  20. { a , b , c , d } P ν 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 1 1 − 1 1 1 0 0 2 2 8 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 1 1 1 0 0 0 0 4 2 8 4 4 4 8 4 { a } { b } { c } { d } Recursive Iterative Rescaling Method M¨ obius transform

  21. { a , b , c , d } P ν 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 1 1 − 1 8 + 1 1 1 0 0 2 2 8 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 8 − 1 1 1 4 − 1 1 1 1 0 − 0 0 0 0 4 2 24 4 12 4 8 4 { a } { b } { c } { d } Rescale Iterative Rescaling Method when negative

  22. { a , b , c , d } P ν 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 1 1 1 1 0 0 0 2 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 1 1 1 0 0 0 0 4 2 12 4 6 4 8 4 { a } { b } { c } { d } Iterative Rescaling Method

  23. { a , b , c , d } P ν 1 1 4 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 1 1 1 1 1 0 0 0 2 2 4 8 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 1 1 1 0 0 0 0 4 2 12 4 6 4 8 4 { a } { b } { c } { d } Recursive Iterative Rescaling Method M¨ obius transform

  24. { a , b , c , d } P P IRM 1 1 1 4 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 11 / 12 5 / 6 1 1 1 1 1 1 1 0 0 0 2 2 2 2 4 4 8 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 / 2 1 / 3 2 / 3 1 / 2 1 1 1 1 1 1 1 1 0 0 0 0 0 0 4 2 2 12 4 4 6 4 4 8 4 4 { a } { b } { c } { d } Iterative Rescaling Method P IRM A = ∑ B ⊆ A ν B M¨ obius inverse

  25. { a , b , c , d } P 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 1 1 1 1 2 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 0 0 2 4 4 4 Minimal { a } { b } { c } { d } Iterative Rescaling Method

  26. { a , b , c , d } P ν 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } 1 1 1 1 2 2 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 1 1 1 0 0 0 0 2 2 4 4 4 4 4 4 Minimal { a } { b } { c } { d } M¨ obius transform Iterative Rescaling Method per cardinality

  27. { a , b , c , d } P ν 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } − 1 1 − 1 1 − 1 1 1 0 2 2 4 2 4 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 1 1 1 1 1 1 1 0 0 0 0 2 2 4 4 4 4 4 4 Minimal { a } { b } { c } { d } M¨ obius transform Iterative Rescaling Method per cardinality

  28. { a , b , c , d } P ν 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } − 1 1 − 1 1 − 1 1 1 0 2 2 4 2 4 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 1 2 − 1 1 1 4 − 1 1 1 1 4 − 1 1 1 0 0 0 0 4 2 8 4 4 4 8 4 Minimal { a } { b } { c } { d } Rescale Iterative Rescaling Method when negative

  29. { a , b , c , d } P ν 1 { a , b , c } { a , b , d } { a , c , d } { b , c , d } − 1 1 − 1 1 − 1 1 1 0 2 2 4 2 4 4 4 { a , b } { a , c } { a , d } { b , c } { b , d } { c , d } 2 − 1 1 1 4 − 1 1 1 4 − 1 1 1 1 4 − 1 1 0 − 0 0 0 0 6 2 8 4 12 4 8 4 Minimal { a } { b } { c } { d } Iterative Rescaling Method Rescale minimally when negative

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