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

Finitary characterizations of sets of lower previsions Erik Quaeghebeur SYSTeMS Research Group Ghent University Belgium

P on K is coherent iff ∑ h ∈ K λ h · Ph ≤ max ∑ h ∈ K λ h · h for all λ in R K with at most one strictly negative component

Pg gc = 1 gb = 2 / 3 ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf λ f · Pf + λ g · Pg ≤ max { λ f , λ f · 1 / 2 + λ g · 2 / 3 , λ g }

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

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

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

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

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

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

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

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

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

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

Pg P c gc = 1 P b gb = 2 / 3 P Ω P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

P ( κ · g + α ) P c gc = κ + α P b gb = κ · 2 / 3 + α P Ω P a ga = α fc = fa = 0 fb = 1 / 2 1 Pf P ( κ · g + α ) = κ · Pg + α for all κ ∈ R ≥ 0 and α ∈ R

Pg P c gc = 1 P b gb = 2 / 3 P Ω P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

P b Pg P a P c P c P Ω gc = 1 P b gb = 2 / 3 P Ω P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

P a Pg Pg = 0 Pf = 1 P c gc = 1 P b gb = 2 / 3 P Ω P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg Pg = 2 / 3 P b P c Pf = 1 / 2 gc = 1 P b gb = 2 / 3 P Ω P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg P c Pg = 1 P c Pf = 0 gc = 1 P b gb = 2 / 3 P Ω P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg = 0 Pg Pf = 0 P Ω P c gc = 1 P b gb = 2 / 3 P Ω P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg P c gc = 1 P b gb = 2 / 3 P Ω P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg P c gc = 1 P b gb = 2 / 3 P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf add I a to K

Pg P c gc = 1 P bc P b gb = 2 / 3 P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf add I c to K

Pg P c gc = 1 P bc P 3 P b gb = 2 / 3 P 2 P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf add I b to K

Pg P c gc = 1 P bc P 3 P b gb = 2 / 3 P 2 P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

P 3 P 2 Pg P b P c gc = 1 P c P a P bc P 3 P b gb = 2 / 3 P Ω P 2 P bc P ab P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg P c P ab gc = 1 P bc P 3 P b gb = 2 / 3 P 2 P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg P bc P c gc = 1 P bc P 3 P b gb = 2 / 3 P 2 P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg P c P 2 gc = 1 P bc P 3 P b gb = 2 / 3 P 2 P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg P 3 P c gc = 1 P bc P 3 P b gb = 2 / 3 P 2 P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg P c gc = 1 P bc P 3 P b gb = 2 / 3 P 2 P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

P on a lattice K is n -monotone iff P is monotone and K | + 1 · P ( � ˇ ˆ K ( − 1 ) | ˇ ˆ P ( � K ) ≥ ∑ K ) for all 1 < k ≤ n and K ⊆ K ˇ ˆ K ⊆

Pg gc = 1 gb = 2 / 3 ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf K lattice based on { f , g , I a , I b , I c } , then project back on R { f , g , I a , I b , I c }

Pg P c gc = 1 P bc P b gb = 2 / 3 P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf complete-monotonicity

Pg P c gc = 1 P bc P b gb = 2 / 3 P ▽ P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf 2 -monotonicity

Pg P c gc = 1 P bc P 3 P b gb = 2 / 3 P 2 P ▽ P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg P c P ab gc = 1 P bc P 3 P b gb = 2 / 3 P 2 P ▽ P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg P bc P c gc = 1 P bc P 3 P b gb = 2 / 3 P 2 P ▽ P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg P c P 2 gc = 1 P bc P 3 P b gb = 2 / 3 P 2 P ▽ P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg P 3 P c gc = 1 P bc P 3 P b gb = 2 / 3 P 2 P ▽ P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg P ▽ P c gc = 1 P bc P 3 P b gb = 2 / 3 P 2 P ▽ P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf

Pg P c gc = 1 P bc P 3 P b gb = 2 / 3 P 2 P ▽ P Ω P ab P a ga = 0 fc = fa = 0 fb = 1 / 2 1 Pf discouraging picture for n -monotone outer approximation accuracy

intentionally left blank

Pg P b gb = 1 gc = 1 / 2 P bc ga = 0 P Ω P a fc = fa = 0 fb = 1 / 2 1 Pf

Pg P a P b gb = 1 gc = 1 / 2 P bc ga = 0 P Ω P a fc = fa = 0 fb = 1 / 2 1 Pf

Pg P b gb = 1 P b gc = 1 / 2 P bc ga = 0 P Ω P a fc = fa = 0 fb = 1 / 2 1 Pf

Pg P b gb = 1 P bc gc = 1 / 2 P bc ga = 0 P Ω P a fc = fa = 0 fb = 1 / 2 1 Pf

Pg P b gb = 1 P Ω gc = 1 / 2 P bc ga = 0 P Ω P a fc = fa = 0 fb = 1 / 2 1 Pf

Pg P b P = gb = 1 P = gc = 1 / 2 P bc ga = 0 P Ω P a fc = fa = 0 fb = 1 / 2 1 Pf f and 1 − f equal up to permutation; impose Pg = Pf

empty on purpose

P ( 1 − f ) 1 − fc = 1 P c 1 − fb = 1 / 2 1 − fa = 0 P Ω P a fc = fa = 0 1 / 2 1 Pf fb =

P a P ( 1 − f ) 1 − fc = 1 P c 1 − fb = 1 / 2 1 − fa = 0 P Ω P a fc = fa = 0 1 / 2 1 Pf fb =

P ( 1 − f ) P Ω 1 − fc = 1 P c 1 − fb = 1 / 2 1 − fa = 0 P Ω P a fc = fa = 0 1 / 2 1 Pf fb =

P ( 1 − f ) 1 − fc = 1 P c P c 1 − fb = 1 / 2 1 − fa = 0 P Ω P a fc = fa = 0 1 / 2 1 Pf fb =

P ( 1 − f ) P = 1 − fc = 1 P c 1 − fb = 1 / 2 P = 1 − fa = 0 P Ω P a fc = fa = 0 1 / 2 1 Pf fb = f and g equal up to permutation; impose P ( 1 − f ) = Pf

Numbers, numbers, numbers Combinatorics for coherent lower probabilities on different K

Numbers, numbers, numbers Combinatorics for coherent lower probabilities on different K Lower pmfs | K | = | Ω | and # λ = # P = | Ω | + 1 (linear-vacuous)

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)

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)

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 ? ?

More numbers, numbers, numbers Combinatorics for n -monotone lower probabilities

More numbers, numbers, numbers Combinatorics for n -monotone lower probabilities Completely monotone | K | = 2 | Ω | , # λ = 2 | Ω | + 3 and # P = 2 | Ω | − 1

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 ?

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

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 · k 2 − 4 · k + 3 )