Why Min-Based Reasonable Properties . . . Conditioning Reasonable - PowerPoint PPT Presentation

Need for Ordinal-Scale . . . From Ordinal Scale to . . . Need for Conditioning . . . Reasonable Properties Why Min-Based Reasonable Properties . . . Conditioning Reasonable Properties . . . Final Property: Invariance Main Result Salem Benferhat 1 and Vladik Kreinovich 2 Proof Home Page 1 CRIL (Centre de Recherche en Informatique de Lens) CNRS – UMR 8188 Title Page Universit´ e d’Artois, Facult´ e des sciences Jean Perrin Rue Jean Souvraz, SP 18, F62307 Lens Cedex, France ◭◭ ◮◮ benferhat@cril.univ-artois.fr ◭ ◮ 2 Department of Computer Science Page 1 of 20 University of Texas at El Paso El Paso, Texas 79968, USA Go Back vladik@utep.edu Full Screen Close Quit

Need for Ordinal-Scale . . . From Ordinal Scale to . . . 1. Need for Ordinal-Scale Possibility Degrees Need for Conditioning . . . • It is often useful to describe, Reasonable Properties Reasonable Properties . . . – for each theoretically possible alternative ω from Reasonable Properties . . . the set of all theoretically possible alternatives Ω, Final Property: Invariance – to what extent this alternative is, in the expert’s Main Result opinion, actually possible. Proof • Often, the only information that we can extract from Home Page experts is the qualitative one: Title Page – which alternatives have a higher degree of possibil- ◭◭ ◮◮ ity and ◭ ◮ – which have lower degree. Page 2 of 20 • In some cases, we have a linear order. Go Back • We could use this order to process this information. Full Screen • However, computers have been designed to process numbers; they are still best in processing numbers. Close Quit

Need for Ordinal-Scale . . . From Ordinal Scale to . . . 2. From Ordinal Scale to Numbers Need for Conditioning . . . • So, degrees of possibility are usually described by num- Reasonable Properties bers π ( ω ) ∈ [0 , 1]: Reasonable Properties . . . Reasonable Properties . . . – the higher the degree, Final Property: Invariance – the larger the value π ( ω ). Main Result • These numbers by themselves do not have an exact Proof meaning, the only meaning is in the order. Home Page • So, the same meaning can be described if we apply any Title Page strictly increasing transformation to [0 , 1]. ◭◭ ◮◮ • Usually, some of this freedom is eliminated by the con- ◭ ◮ vention that the largest degree is set to 1. Page 3 of 20 • We can always achieve this with an appropriate trans- Go Back formation ( normalization ). Full Screen • Definition. Let Ω be a finite set. A possibility distri- bution is a function π : Ω → [0 , 1] s.t. max ω ∈ Ω π ( ω ) = 1 . Close Quit

Need for Ordinal-Scale . . . From Ordinal Scale to . . . 3. Need for Conditioning and Normalization Need for Conditioning . . . • Often, we acquire an additional information: Reasonable Properties Reasonable Properties . . . – some of the alternatives that we originally thought Reasonable Properties . . . to be possible Final Property: Invariance – are actually not possible: Ψ ⊂ Ω, Ψ � = Ω. Main Result • Example: some original suspects have alibis. Proof • We have π ′ ( ω ) = 0 for all ω �∈ Ψ; but we may have Home Page ω ∈ Ψ π ′ ( ω ) < 1, so we need normalization. max Title Page • Definition. By a conditioning operator , we mean a ◭◭ ◮◮ mapping ( π | Ψ) that: ◭ ◮ – inputs a possibility distribution π on a set Ω and a Page 4 of 20 non-empty set Ψ ⊆ Ω , and Go Back – returns a new possibility distribution for which ( π | Ψ)( ω ) = 0 for all ω �∈ Ψ . Full Screen • What are the reasonable conditioning operators? Close Quit

Need for Ordinal-Scale . . . From Ordinal Scale to . . . 4. Reasonable Properties Need for Conditioning . . . • A first reasonable requirement is that: Reasonable Properties Reasonable Properties . . . – since alternatives ω �∈ Ψ are excluded, Reasonable Properties . . . – their original possibility degrees should not affect Final Property: Invariance the resulting degrees. Main Result • C1. If π | Ψ = π ′ | Ψ , i.e., if π ( ω ) = π ′ ( ω ) for all ω ∈ Ψ, Proof then ( π | Ψ) = ( π ′ | Ψ) . Home Page • Another reasonable condition is that: Title Page – while the numerical values of possibility degrees ◭◭ ◮◮ may change, ◭ ◮ – the order between these degrees should not change. Page 5 of 20 If π ( ω ) < π ( ω ′ ) for some ω, ω ′ ∈ Ψ, then • C2. ( π | Ψ)( ω ) < ( π | Ψ)( ω ′ ) . Go Back If π ( ω ) = π ( ω ′ ) for some ω, ω ′ ∈ Ψ, then Full Screen • C3. ( π | Ψ)( ω ) = ( π | Ψ)( ω ′ ) . Close Quit

Need for Ordinal-Scale . . . From Ordinal Scale to . . . 5. Reasonable Properties (cont-d) Need for Conditioning . . . • Often, after learning Ψ ⊂ Ω, we learn additional infor- Reasonable Properties mation Ψ ′ ⊂ Ψ. In this case: Reasonable Properties . . . – first compute π ′ = ( π | Ψ), and then Reasonable Properties . . . Final Property: Invariance – compute π ′′ = ( π ′ | Ψ ′ ) = (( π | Ψ) | Ψ ′ ). Main Result • Alternative, we could learn both pieces of the informa- Proof tion at the same time, and get ( π | Ψ ′ ). Home Page • In both cases, we gain the exact same new information. Title Page ◭◭ ◮◮ • So, the resulting changes in possibility degrees should be the same: ◭ ◮ • C4. If Ψ ′ ⊂ Ψ, then (( π | Ψ) | Ψ ′ ) = ( π | Ψ ′ ) . Page 6 of 20 Go Back Full Screen Close Quit

Need for Ordinal-Scale . . . From Ordinal Scale to . . . 6. Reasonable Properties (cont-d) Need for Conditioning . . . • Another condition is that if had an alternative ω 0 which Reasonable Properties we originally believed to be impossible, then: Reasonable Properties . . . Reasonable Properties . . . – this alternative should remain impossible, and Final Property: Invariance – the possibility degrees of all other alternatives ω � = Main Result ω 0 should remain the same. Proof • C5. If π ( ω 0 ) = 0 for some ω 0 ∈ Ψ, then ( π | Ψ)( ω 0 ) = 0 Home Page and ( π | Ψ −{ ω 0 } | Ψ) = ( π | Ψ) | Ψ −{ ω 0 } . Title Page ◭◭ ◮◮ ◭ ◮ Page 7 of 20 Go Back Full Screen Close Quit

Need for Ordinal-Scale . . . From Ordinal Scale to . . . 7. Final Property: Invariance Need for Conditioning . . . • What matters is the order between the degrees, not the Reasonable Properties numerical values of the degrees. Reasonable Properties . . . Reasonable Properties . . . • So, the situations should not change if we apply a re- Final Property: Invariance scaling T that doesn’t change the order (e.g., x → x 2 ). Main Result • The result of applying the conditioning operator not Proof change if we apply such a re-scaling. Home Page • We should get the exact same result: Title Page – if we apply conditioning π → ( π | Ψ) in the original ◭◭ ◮◮ scale, and then re-scale to T ( π | Ψ); ◭ ◮ – or we first apply the re-scaling, resulting in Tπ , and Page 8 of 20 then apply the conditioning, resulting in ( Tπ | Ψ). Go Back • C6. For every increasing one-to-one function Full Screen T : [0 , 1] → [0 , 1], we have ( Tπ | Ψ) = T ( π | Ψ) . Close Quit

Need for Ordinal-Scale . . . From Ordinal Scale to . . . 8. Main Result Need for Conditioning . . . • Proposition. The only conditioning operator satisfying Reasonable Properties C1 – C6 is the min-based operator for which: Reasonable Properties . . . Reasonable Properties . . . ω ′ ∈ Ψ π ( ω ′ ) ; • ( π | Ψ)( ω ) = 1 when ω ∈ Ω and π ( ω ) = max Final Property: Invariance • ( π | Ψ)( ω ) = π ( ω ) when ω ∈ Ω and Main Result ω ′ ∈ Ψ π ( ω ′ ) ; and π ( ω ) < max Proof Home Page • ( π | Ψ)( ω ) = 0 when ω �∈ Ψ . Title Page • The usual derivation selects ( A | B ) as the maximal value s.t. d ( A & B ) = d (( A | B ) & B ), with ◭◭ ◮◮ ◭ ◮ def d ( A & B ) = min( d ( A ) , d ( B )) . Page 9 of 20 • We show that maximality can be replaced with invari- Go Back ance – reflecting ordinal character of degrees. Full Screen Close Quit

Need for Ordinal-Scale . . . From Ordinal Scale to . . . 9. Proof Need for Conditioning . . . • It is easy to show that the min-based operator satisfies Reasonable Properties the properties C1 – C6 . Reasonable Properties . . . Reasonable Properties . . . • To complete the proof, we need to prove that, vice Final Property: Invariance versa, Main Result – every conditioning operator that satisfies these five Proof properties Home Page – is indeed the min-based operator. Title Page • To prove this statement, we will consider two possible ◭◭ ◮◮ cases: ◭ ◮ – the case when the set Ψ contains some alternative Page 10 of 20 ω for which π ( ω ) = 1, and Go Back – the case when the set Ψ does not contain any al- ternative ω for which π ( ω ) = 1. Full Screen Close Quit