Independent Natural Extension for Infinite Spaces Williams-coherence to the Rescue! Jasper De Bock Ghent University Belgium
X 1 X 2 independent local local uncertainty uncertainty model model ? joint uncertainty model
P ( X 1 | X 2 ) = P ( X 1 ) P ( X 2 | X 1 ) = P ( X 2 ) X 1 X 2 independent local local uncertainty uncertainty model model P ( X 1 ) P ( X 2 ) ? joint uncertainty model P ( X 1 , X 2 )
P ( X 1 | X 2 ) = P ( X 1 ) P ( X 2 | X 1 ) = P ( X 2 ) X 1 X 2 independent local local uncertainty uncertainty model model P ( X 1 ) P ( X 2 ) joint uncertainty model P ( X 1 , X 2 ) = P ( X 1 ) P ( X 2 )
X 1 local uncertainty model P ( X 1 ) P ( X 1 ) P ( f ( X 1 )) P ( f ( X 1 )) D 1 , , , , , . . . E ( f ( X 1 )) E ( f ( X 1 ))
X 1 local uncertainty model P ( X 1 ) P ( X 1 ) P ( f ( X 1 )) P ( f ( X 1 )) D 1 , , , , , . . . E ( f ( X 1 )) E ( f ( X 1 ))
X 1 X 2 independent local local uncertainty uncertainty model model P ( f ( X 1 )) P ( f ( X 2 )) ? joint uncertainty model P ( f ( X 1 , X 2 ))
? X 1 X 2 independent local local uncertainty uncertainty model model P ( f ( X 1 )) P ( f ( X 2 )) ? joint uncertainty model P ( f ( X 1 , X 2 ))
P ( f ( X 1 ) | X 2 ) = P ( f ( X 1 )) P ( f ( X 2 ) | X 1 ) = P ( f ( X 2 )) X 1 X 2 independent local local uncertainty uncertainty model model P ( f ( X 1 )) P ( f ( X 2 )) ? joint uncertainty model P ( f ( X 1 , X 2 ))
P ( f ( X 1 ) | X 2 ) = P ( f ( X 1 )) P ( f ( X 2 ) | X 1 ) = P ( f ( X 2 )) X 1 X 2 independent local local uncertainty uncertainty Coherence model model P ( f ( X 1 )) P ( f ( X 2 )) ? joint uncertainty model P ( f ( X 1 , X 2 ))
P ( f ( X 1 ) | X 2 ) = P ( f ( X 1 )) P ( f ( X 2 ) | X 1 ) = P ( f ( X 2 )) X 1 X 2 independent local local uncertainty uncertainty Coherence model model P ( f ( X 1 )) P ( f ( X 2 )) = = P 2 ( f ) P 1 ( f ) joint uncertainty model ( P 1 ⊗ P 2 )( f ( X 1 , X 2 ))
Independent Natural Extension for Infinite Spaces Williams-coherence to the Rescue! Jasper De Bock Ghent University Belgium
Two very useful properties External additivity ( P 1 ⊗ P 2 )( f ( X 1 ) + h ( X 2 )) = P 1 ( f ( X 1 )) + P 2 ( h ( X 2 )) Factorisation ( P 1 ⊗ P 2 )( g ( X 1 ) h ( X 2 )) ( P 1 ( g ( X 1 )) P 2 ( h ( X 2 )) if P ( h ( X 2 )) ≥ 0 = P 1 ( g ( X 1 )) P 2 ( h ( X 2 )) if P ( h ( X 2 )) ≤ 0 if g ≥ 0
DISCLAIMER! All of this is well known, and has been for several years now… de Cooman, Miranda & Zaffalon 2011 de Cooman & Miranda 2012
DISCLAIMER! All of this is well known, and has been for several years now… de Cooman, Miranda & …but Zaffalon 2011 de Cooman & only for Miranda 2012 finite spaces!
Independent Natural Extension for Infinite Spaces ?
P ( f ( X 1 ) | X 2 ) = P ( f ( X 1 )) P ( f ( X 2 ) | X 1 ) = P ( f ( X 2 )) X 1 X 2 independent local local uncertainty uncertainty Coherence model model ? P ( f ( X 1 )) P ( f ( X 2 )) ? joint uncertainty model
Coherence ? Walley Williams
Independent natural extension may not exist! Miranda & Zaffalon 2015 Coherence ? Walley Williams
Independent Independent natural extension natural extension may not exist! always exists! De Bock 2017 Miranda & Vicig Zaffalon 2015 Coherence 2000 Walley Williams
Independent Natural Extension for Infinite Spaces Williams-coherence to the Rescue! !
Two very useful properties ? External additivity ( P 1 ⊗ P 2 )( f ( X 1 ) + h ( X 2 )) = P 1 ( f ( X 1 )) + P 2 ( h ( X 2 )) ? Factorisation ( P 1 ⊗ P 2 )( g ( X 1 ) h ( X 2 )) ( P 1 ( g ( X 1 )) P 2 ( h ( X 2 )) if P ( h ( X 2 )) ≥ 0 = P 1 ( g ( X 1 )) P 2 ( h ( X 2 )) if P ( h ( X 2 )) ≤ 0 if g ≥ 0
Two very useful properties External additivity ( P 1 ⊗ P 2 )( f ( X 1 ) + h ( X 2 )) = P 1 ( f ( X 1 )) + P 2 ( h ( X 2 )) ? Factorisation ( P 1 ⊗ P 2 )( g ( X 1 ) h ( X 2 )) ( P 1 ( g ( X 1 )) P 2 ( h ( X 2 )) if P ( h ( X 2 )) ≥ 0 = P 1 ( g ( X 1 )) P 2 ( h ( X 2 )) if P ( h ( X 2 )) ≤ 0 if g ≥ 0
P ( f ( X 1 ) | X 2 ) = P ( f ( X 1 )) P ( f ( X 2 ) | X 1 ) = P ( f ( X 2 )) X 1 X 2 independent local local uncertainty uncertainty Coherence model model P ( f ( X 1 )) P ( f ( X 2 )) = = P 2 ( f ) P 1 ( f ) joint uncertainty model ( P 1 ⊗ P 2 )( f ( X 1 , X 2 ))
P ( f ( X 1 ) | X 2 ) = P ( f ( X 1 )) P ( f ( X 2 ) | X 1 ) = P ( f ( X 2 )) independent P ( f ( X 1 ) | B 2 ) = P ( f ( X 1 )) ∀ B 2 ∈ B 2 P ( f ( X 2 ) | B 1 ) = P ( f ( X 2 )) ∀ B 1 ∈ B 1
P ( f ( X 1 ) | X 2 ) = P ( f ( X 1 )) P ( f ( X 2 ) | X 1 ) = P ( f ( X 2 )) independent P ( f ( X 1 ) | B 2 ) = P ( f ( X 1 )) ∀ B 2 ∈ B 2 P ( f ( X 2 ) | B 1 ) = P ( f ( X 2 )) ∀ B 1 ∈ B 1 value-independence: B i = {{ x i } : x i ∈ X i } subset-independence: B i = P ( X i ) \ { ∅ }
Two very useful properties External additivity ( P 1 ⊗ P 2 )( f ( X 1 ) + h ( X 2 )) = P 1 ( f ( X 1 )) + P 2 ( h ( X 2 )) ? Factorisation ( P 1 ⊗ P 2 )( g ( X 1 ) h ( X 2 )) ( P 1 ( g ( X 1 )) P 2 ( h ( X 2 )) if P ( h ( X 2 )) ≥ 0 = P 1 ( g ( X 1 )) P 2 ( h ( X 2 )) if P ( h ( X 2 )) ≤ 0 if g ≥ 0
Two very useful properties External additivity ( P 1 ⊗ P 2 )( f ( X 1 ) + h ( X 2 )) = P 1 ( f ( X 1 )) + P 2 ( h ( X 2 )) Factorisation ( P 1 ⊗ P 2 )( g ( X 1 ) h ( X 2 )) ( P 1 ( g ( X 1 )) P 2 ( h ( X 2 )) if P ( h ( X 2 )) ≥ 0 = P 1 ( g ( X 1 )) P 2 ( h ( X 2 )) if P ( h ( X 2 )) ≤ 0 if g ≥ 0 is B 1 -measurable
Independent Independent natural extension natural extension may not exist! always exists! Walley Williams value- subset- independence independence Factorisation Factorisation may not hold! always holds!
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