Finitary characterizations of sets of lower previsions Erik - - PowerPoint PPT Presentation

▶
finitary characterizations of sets of lower previsions
SMART_READER_LITE
LIVE PREVIEW

Finitary characterizations of sets of lower previsions Erik - - PowerPoint PPT Presentation

Mar 21, 2023 190 likes •911 views

Finitary characterizations of sets of lower previsions Erik Quaeghebeur SYSTeMS Research Group Ghent University Belgium P on K is coherent iff h K h Ph max h K h h for all in R K with at most one strictly

slide-1
SLIDE 1

Finitary characterizations of sets of lower previsions

Erik Quaeghebeur

SYSTeMS Research Group Ghent University Belgium

slide-2
SLIDE 2

P on K is coherent iff ∑h∈K λh ·Ph ≤ max∑h∈K λh ·h

for all λ in RK with at most one strictly negative component

slide-3
SLIDE 3

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 λf ·Pf +λg ·Pg ≤ max{λf ,λf · 1/2+λg · 2/3,λg}

slide-4
SLIDE 4

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 (−1,0) −Pf ≤ max{−1,−1/2,0} = 0

slide-5
SLIDE 5

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 (−1,0) (0,−1)

slide-6
SLIDE 6

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 (−1,0) (0,−1) (1,0)

slide-7
SLIDE 7

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 (−1,0) (0,−1) (1,0) (0,1)

slide-8
SLIDE 8

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 (−1,0) (0,−1) (1,0) (0,1) (−1,1)

slide-9
SLIDE 9

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 (−1,0) (0,−1) (1,0) (0,1) (−1,1) (1,−1)

slide-10
SLIDE 10

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 (−1,0) (0,−1) (1,0) (0,1) (1,−1) (1, 3/4)

slide-11
SLIDE 11

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 (−1,0) (0,−1) (1,0) (0,1) (1, 3/4) (2/3,1)

slide-12
SLIDE 12

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 (−1,0) (0,−1) (1,0) (0,1) (1, 3/4) (2/3,1)

slide-13
SLIDE 13

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 (−1,0) (0,−1) (1, 3/4) (2/3,1)

slide-14
SLIDE 14

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ

slide-15
SLIDE 15

Pf fc =

1/2

fb = 1 fa = P(κ ·g+α) ga = α gb = κ · 2/3+α gc = κ +α Pc Pb Pa PΩ P(κ ·g+α) = κ ·Pg+α for all κ ∈ R≥0 and α ∈ R

slide-16
SLIDE 16

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ

slide-17
SLIDE 17

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ PΩ Pa Pc Pb

slide-18
SLIDE 18

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pf = 1 Pg = 0 Pa

slide-19
SLIDE 19

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pf = 1/2 Pg = 2/3 Pb

slide-20
SLIDE 20

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pf = 0 Pg = 1 Pc

slide-21
SLIDE 21

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ PΩ Pf = 0 Pg = 0

slide-22
SLIDE 22

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ

slide-23
SLIDE 23

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab

add Ia to K

slide-24
SLIDE 24

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc

add Ic to K

slide-25
SLIDE 25

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3

add Ib to K

slide-26
SLIDE 26

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3

slide-27
SLIDE 27

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3 PΩ Pab Pbc Pa Pc Pb P2 P3

slide-28
SLIDE 28

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3 Pab

slide-29
SLIDE 29

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3 Pbc

slide-30
SLIDE 30

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3 P2

slide-31
SLIDE 31

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3 P3

slide-32
SLIDE 32

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3

slide-33
SLIDE 33

P on a lattice K is n-monotone iff P is monotone and P( ˆ K ) ≥ ∑

ˇ K ⊆ ˆ K (−1)| ˇ K |+1 ·P(

ˇ K ) for all 1 < k ≤ n and ˆ K ⊆ K

slide-34
SLIDE 34

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 K lattice based on {f,g,Ia,Ib,Ic}, then project back on R{f,g,Ia,Ib,Ic}

slide-35
SLIDE 35

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc

complete-monotonicity

slide-36
SLIDE 36

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P▽ 2-monotonicity

slide-37
SLIDE 37

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3 P▽

slide-38
SLIDE 38

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3 P▽ Pab

slide-39
SLIDE 39

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3 P▽ Pbc

slide-40
SLIDE 40

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3 P▽ P2

slide-41
SLIDE 41

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3 P▽ P3

slide-42
SLIDE 42

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3 P▽ P▽

slide-43
SLIDE 43

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 2/3 gc = 1 Pc Pb Pa PΩ Pab Pbc P2 P3 P▽

discouraging picture for n-monotone outer approximation accuracy

slide-44
SLIDE 44

intentionally left blank

slide-45
SLIDE 45

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 1 gc = 1/2 Pb Pa PΩ Pbc

slide-46
SLIDE 46

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 1 gc = 1/2 Pb Pa PΩ Pbc Pa

slide-47
SLIDE 47

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 1 gc = 1/2 Pb Pa PΩ Pbc Pb

slide-48
SLIDE 48

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 1 gc = 1/2 Pb Pa PΩ Pbc Pbc

slide-49
SLIDE 49

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 1 gc = 1/2 Pb Pa PΩ Pbc PΩ

slide-50
SLIDE 50

Pf fc =

1/2

fb = 1 fa = Pg ga = 0 gb = 1 gc = 1/2 Pb Pa PΩ Pbc P= P= f and 1−f equal up to permutation; impose Pg = Pf

slide-51
SLIDE 51

empty on purpose

slide-52
SLIDE 52

Pf fc =

1/2

fb = 1 fa = P(1−f) 1−fa = 0 1−fc = 1 1−fb = 1/2 Pc Pa PΩ

slide-53
SLIDE 53

Pf fc =

1/2

fb = 1 fa = P(1−f) 1−fa = 0 1−fc = 1 1−fb = 1/2 Pc Pa PΩ Pa

slide-54
SLIDE 54

Pf fc =

1/2

fb = 1 fa = P(1−f) 1−fa = 0 1−fc = 1 1−fb = 1/2 Pc Pa PΩ PΩ

slide-55
SLIDE 55

Pf fc =

1/2

fb = 1 fa = P(1−f) 1−fa = 0 1−fc = 1 1−fb = 1/2 Pc Pa PΩ Pc

slide-56
SLIDE 56

Pf fc =

1/2

fb = 1 fa = P(1−f) 1−fa = 0 1−fc = 1 1−fb = 1/2 Pc Pa PΩ P= P= f and g equal up to permutation; impose P(1−f) = Pf

slide-57
SLIDE 57

Numbers, numbers, numbers

Combinatorics for coherent lower probabilities on different K

slide-58
SLIDE 58

Numbers, numbers, numbers

Combinatorics for coherent lower probabilities on different K

Lower pmfs |K | = |Ω| and #λ = #P = |Ω|+1 (linear-vacuous)

slide-59
SLIDE 59

Numbers, numbers, numbers

Combinatorics for coherent lower probabilities on different K

Lower pmfs |K | = |Ω| and #λ = #P = |Ω|+1 (linear-vacuous) Upper pmfs |K | = |Ω|, #λ = 2·|Ω|+1 and #P = 2|Ω| +1 (not completely monotone)

slide-60
SLIDE 60

Numbers, numbers, numbers

Combinatorics for coherent lower probabilities on different K

Lower pmfs |K | = |Ω| and #λ = #P = |Ω|+1 (linear-vacuous) Upper pmfs |K | = |Ω|, #λ = 2·|Ω|+1 and #P = 2|Ω| +1 (not completely monotone) Probability intervals |K | = 2·|Ω| for |Ω| > 2 and

|Ω| 2 3 4 5 6 7 8 9 10 #λ 3 9 16 20 24 28 32 36 40 #P 3 8 20 47 105 226 474 977 1991

(subset of 2-monotone)

slide-61
SLIDE 61

Numbers, numbers, numbers

Combinatorics for coherent lower probabilities on different K

Lower pmfs |K | = |Ω| and #λ = #P = |Ω|+1 (linear-vacuous) Upper pmfs |K | = |Ω|, #λ = 2·|Ω|+1 and #P = 2|Ω| +1 (not completely monotone) Probability intervals |K | = 2·|Ω| for |Ω| > 2 and

|Ω| 2 3 4 5 6 7 8 9 10 #λ 3 9 16 20 24 28 32 36 40 #P 3 8 20 47 105 226 474 977 1991

(subset of 2-monotone) Lower probabilities |K | = 2|Ω| and

|Ω| 2 3 4 5 6 #λ 3 (3) 9 (17) 48 (179) 285 (7351) ? (?) #P 3 8 402 ? ?

slide-62
SLIDE 62

More numbers, numbers, numbers

Combinatorics for n-monotone lower probabilities

slide-63
SLIDE 63

More numbers, numbers, numbers

Combinatorics for n-monotone lower probabilities

Completely monotone |K | = 2|Ω|, #λ = 2|Ω| +3 and #P = 2|Ω| −1

slide-64
SLIDE 64

More numbers, numbers, numbers

Combinatorics for n-monotone lower probabilities

Completely monotone |K | = 2|Ω|, #λ = 2|Ω| +3 and #P = 2|Ω| −1

2-monotone |K | = 2|Ω| and |Ω| 2 3 4 5 6 #λ 7 (10) 13 (32) 32 (124) 89 (500) ? (?) #P 3 8 41 117983 ?

slide-65
SLIDE 65

Still more numbers, numbers, numbers

Combinatorics for coherent lower previsions on different K

Consider K consisting of gambles taking values in

  • ℓ/k : 0 ≤ ℓ ≤ k
slide-66
SLIDE 66

Still more numbers, numbers, numbers

Combinatorics for coherent lower previsions on different K

Consider K consisting of gambles taking values in

  • ℓ/k : 0 ≤ ℓ ≤ k
  • |Ω| = 3 |K | = 2·k ·|Ω|, #λ = (2·k +1)·|Ω|, and

#P = (3·k +1)·(3·k2 −4·k +3)

slide-67
SLIDE 67

Still more numbers, numbers, numbers

Combinatorics for coherent lower previsions on different K

Consider K consisting of gambles taking values in

  • ℓ/k : 0 ≤ ℓ ≤ k
  • |Ω| = 3 |K | = 2·k ·|Ω|, #λ = (2·k +1)·|Ω|, and

#P = (3·k +1)·(3·k2 −4·k +3) |Ω| = 4 computationally too demanding

slide-68
SLIDE 68

Conclusion & To Do

slide-69
SLIDE 69

Conclusion & To Do

◮ We can get a view of (projected) sets of lower previsions for

◮ building intuition; ◮ pedagogical use; ◮ more practical applications?

slide-70
SLIDE 70

Conclusion & To Do

◮ We can get a view of (projected) sets of lower previsions for

◮ building intuition; ◮ pedagogical use; ◮ more practical applications?

◮ For small |Ω| or |K | we can

◮ efficiently check coherence for multiple lower previsions; ◮ obtain the extreme lower previsions.

slide-71
SLIDE 71

Conclusion & To Do

◮ We can get a view of (projected) sets of lower previsions for

◮ building intuition; ◮ pedagogical use; ◮ more practical applications?

◮ For small |Ω| or |K | we can

◮ efficiently check coherence for multiple lower previsions; ◮ obtain the extreme lower previsions.

◮ Expand scope by adding contingent gambles?

(for looking at independence concepts)