Completely monotone outer approximations of lower probabilities on - - PowerPoint PPT Presentation

Completely monotone outer approximations

• f

lower probabilities

• n

finite possibility spaces

Erik Quaeghebeur

SYSTeMS Research Group Ghent University Belgium

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} / {a} {b} {c} {d} {a,b} {c,d}

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} / {a} {b} {c} {d} {a,b} {c,d} P 1

5 8 5 8 1 2 1 2 1 2 3 8 3 8 3 8 1 4 1 4 1 8 1 8 1 8 1 8

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} / {a} {b} {c} {d} {a,b} {c,d} P = 1

2 ·P+ 1 2 ·R

1 1 1

5 8 3 4 1 2 5 8 3 4 1 2 1 2 3 4 1 4 1 2 3 4 1 4 1 2 1 2 1 2 3 8 1 2 1 4 3 8 1 2 1 4 3 8 1 2 1 4 1 4 1 2 0 1 4 1 2 0 1 8 1 4 0 1 8 1 4 0 1 8 1 4 0 1 8 1 4 0

0 0 0

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a} {b} {c} {d} {a,b} {c,d} P = 1

2 ·P+ 1 2 ·R

EP = 1

2 ·Ep + 1 2 ·ER

linear–imprecise decomposition

1

1 2 1 2 1 4 1 4 1 2 1 4 1 4 1 4 1 4 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P 1

1 2 1 2 1 4 1 4 1 2 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P µA = PA−∑B⊂A µB

Recursive M¨

• bius transform

1

1 2 1 2 1 4 1 4 1 2 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P µA = PA−∑B⊂A µB

Recursive M¨

• bius transform

µ 1

1 2 1 2 1 4 1 4 1 2 1 2 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P µA = PA−∑B⊂A µB

Recursive M¨

• bius transform

µ 1

1 2 1 2 1 4 1 4 1 2 1 2 1 4 1 4 1 4 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P µA = PA−∑B⊂A µB

Recursive M¨

• bius transform

µ 1

1 2 1 2 1 4 1 4 1 2 1 2 1 4 1 4 1 4 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P µA = PA−∑B⊂A µB

Recursive M¨

• bius transform

µ 1

1 2

−1

2 1 2 1 4 1 4 1 2 1 2 1 4 1 4 1 4 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P µA = PA−∑B⊂A µB

Recursive M¨

• bius transform

µ 1

1 2

−1

2 1 2

−1

4 1 4

−1

4 1 4 1 2 1 2 1 4 1 4 1 4 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P µA = PA−∑B⊂A µB

Recursive M¨

• bius transform

µ 1

1 2

−1

2 1 2

−1

4 1 4

−1

4 1 4 1 2 1 2 1 4 1 4 1 4 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P µA = PA−∑B⊂A µB

Recursive M¨

• bius transform

µ 1

3 4 1 2

−1

2 1 2

−1

4 1 4

−1

4 1 4 1 2 1 2 1 4 1 4 1 4 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P µA = PA−∑B⊂A µB

Recursive M¨

• bius transform

µ PA = ∑B⊆A µB M¨

• bius inverse

1

3 4 1 2

−1

2 1 2

−1

4 1 4

−1

4 1 4 1 2 1 2 1 4 1 4 1 4 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Iterative Rescaling Method

1

1 2 1 2 1 4 1 4 1 2 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Iterative Rescaling Method

Recursive M¨

• bius transform

ν 1

1 2

−1

2 1 2 1 4 1 4 1 2 1 2 1 4 1 4 1 4 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Iterative Rescaling Method

Rescale when negative

ν 1

1 2

−1

2 +1 2 1 2 1 4 1 4 1 2 1 2 −1 4 1 4 1 4 −1 8 1 4 1 4 1 4 1 4 −1 8

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Iterative Rescaling Method

ν 1

1 2 1 2 1 4 1 4 1 2 1 4 1 4 1 8 1 4 1 4 1 4 1 8

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Iterative Rescaling Method

Recursive M¨

• bius transform

ν 1

1 2 1 2 1 4

−1

8 1 4 1 2 1 4 1 4 1 8 1 4 1 4 1 4 1 8

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Iterative Rescaling Method

Rescale when negative

ν 1

1 2 1 2 1 4

−1

8 +1 8 1 4 1 2 1 4 1 4 1 8− 1 24 1 4 1 4− 1 12 1 4 1 8

0−0

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Iterative Rescaling Method

ν 1

1 2 1 2 1 4 1 4 1 2 1 4 1 4 1 12 1 4 1 6 1 4 1 8

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Iterative Rescaling Method

Recursive M¨

• bius transform

ν 1

1 4 1 2 1 2 1 4 1 4 1 8 1 2 1 4 1 4 1 12 1 4 1 6 1 4 1 8

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Iterative Rescaling Method

PIRM PIRMA = ∑B⊆A νB M¨

• bius inverse

1 1

1 4 1 2

11/12

2 1 2

5/6

2 1 4 1 4 1 4 1 8 1 4 1 2 1 4

1/2

2 1 4 1 12

1/3

4 1 4 1 6

2/3

4 1 4 1 8

1/2

4

0 0 0 0

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Iterative Rescaling Method Minimal

1

1 2 1 2 1 4 1 4 1 2 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P ν

Iterative Rescaling Method

M¨

• bius transform

per cardinality

Minimal

1

1 2 1 2 1 4 1 4 1 2 1 2 1 4 1 4 1 4 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P ν

Iterative Rescaling Method

M¨

• bius transform

per cardinality

Minimal

1

1 2

−1

2 1 2

−1

4 1 4

−1

4 1 4 1 2 1 2 1 4 1 4 1 4 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Iterative Rescaling Method

Rescale when negative

ν

Minimal

1

1 2

−1

2 1 2

−1

4 1 4

−1

4 1 4 1 2 1 2 −1 4 1 4 1 4 −1 8 1 4 1 4 1 4 1 4 −1 8

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P ν

Iterative Rescaling Method Rescale minimally

when negative

Minimal

1

1 2

−1

2 1 2

−1

4 1 4

−1

4 1 4 1 2 1 2 −1 6 1 4 1 4 −1 8 1 4 1 4− 1 12 1 4 1 4 −1 8

0−0

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P ν

Iterative Rescaling Method Rescale minimally

when negative

Minimal

1

1 2

−1

2 1 2

−1

4 1 4

−1

4 1 4 1 2 1 2 −1 6 1 4 1 4 −1 8 1 4 1 4− 1 12 1 4 1 4 −1 8

0−0 0−0

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Iterative Rescaling Method

ν

Minimal

1

1 2 1 2 1 4 1 4 1 2 1 3 1 4 1 8 1 4 1 6 1 4 1 8

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P ν

Iterative Rescaling Method

M¨

• bius transform

per cardinality

Minimal

1

1 2

− 1

12 1 2 1 4

− 1

24 1 4 1 8 1 2 1 3 1 4 1 8 1 4 1 6 1 4 1 8

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P ν

Iterative Rescaling Method Rescale minimally

when negative

Minimal

1

1 2

− 1

12 1 2 1 4

− 1

24 1 4 1 8 1 2 1 3− 1 21 1 4 1 8− 1 56 1 4 1 6− 1 42 1 4 1 8− 1 56

0−0

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Iterative Rescaling Method

ν

Minimal

1

1 2 1 2 1 4 1 4 1 2 2 7 1 4 3 28 1 4 1 7 1 4 3 28

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P ν

Iterative Rescaling Method

M¨

• bius transform

per cardinality

Minimal

1

1 2 1 2 1 14 1 4 1 4 1 7 1 2 2 7 1 4 3 28 1 4 1 7 1 4 3 28

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P ν

Iterative Rescaling Method

M¨

• bius transform

per cardinality

Minimal

1

1 7 1 2 1 2 1 14 1 4 1 4 1 7 1 2 2 7 1 4 3 28 1 4 1 7 1 4 3 28

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Iterative Rescaling Method Minimal

PIMRM PIMRMA = ∑B⊆A νB M¨

• bius inverse

1 1

1 7 1 2 1 2 1 2 1 14 1 2 1 4 1 4 1 4 1 7 1 4 1 2 2 7

4/7

2 1 4 3 28

3/7

4 1 4 1 7

4/7

4 1 4 3 28

3/7

4

0 0 0 0

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Optimization Approach

1

1 2 1 2 1 4 1 4 1 2 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Linear Programming

PLP

minimize ∑A⊆Ω|PA−PLPA| subject to PLP ≤ P PLP is ∞-monotone

1 1

1 2 1 2 1 2 1 2 1 4 1 4 1 4 1 4 1 2

1/2

2 1 4 4 1 4 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P ν

Linear Programming

minimize ∑A⊆Ω|PA−PLPA| subject to PLP ≤ P PLP is ∞-monotone maximize ∑B⊆Ω 2|Ω\B|νB subject to ∀A⊆Ω(∑B⊆A νB ≤ PA) ν ≥ 0, ∑B⊆Ω νB = 1

1 1

1 4 1 2 1 2 1 2 1 2 1 4 1 4 1 4 1 4 1 2 1 4

1/2

2 1 4 4 1 4 1 4 1 4 1 4 1 4 1 4

0 0 0 0

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Probability Interval Approach

1

1 2 1 2 1 4 1 4 1 2 1 4 1 4 1 4

{a,b,c,d} {a,b,c} {a,c} {a} {a,b,d} {a,d} {b} {a,c,d} {b,c} {c} {b,c,d} {b,d} {d} {a,b} {c,d} P

Probability Interval Approach

PPI 1 1

1 2 1 2 1 2 1 2 1 4 1 4 1 4 1 4 1 2 2 1 4 4 1 4 4 1 4 4

Conclusions

From the paper:

◮ linear-imprecise decomposition is nice ◮ IMRM bests IRM at increased computational cost ◮ IMRM and LP have different strengths ◮ (PI is just lousy)

For the future:

◮ non-LP optimization approaches? ◮ generalize idea IMRM to less-than-complete monotonicity?