Definitions Outline Definitions: Radon-Nikodym derivative Radon-Nikodym derivative: (additive measures) • Concept: ν absolutely continuous w.r.t. µ (if µ ( A ) = 0 then ν ( A ) = 0 ) • Theorem. µ and ν two additive measures on (Ω , F ) and µ be σ -finite. If ν << µ , then there exists a nonnegative measurable function f on Ω such that � ν ( A ) = fdµ A → this permits to define the Radon-Nikodym derivative: Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 18 / 70
Definitions Outline Definitions: Radon-Nikodym derivative Radon-Nikodym derivative: (additive measures) • Concept: ν absolutely continuous w.r.t. µ (if µ ( A ) = 0 then ν ( A ) = 0 ) • Theorem. µ and ν two additive measures on (Ω , F ) and µ be σ -finite. If ν << µ , then there exists a nonnegative measurable function f on Ω such that � ν ( A ) = fdµ A → this permits to define the Radon-Nikodym derivative: → The function f is called the Radon-Nikodym derivative of ν w.r.t. µ , denoted f = dν dµ. Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 18 / 70
Definitions Outline Definitions: Radon-Nikodym derivative Radon-Nikodym derivative: (additive measures) • Concept: ν absolutely continuous w.r.t. µ (if µ ( A ) = 0 then ν ( A ) = 0 ) • Theorem. µ and ν two additive measures on (Ω , F ) and µ be σ -finite. If ν << µ , then there exists a nonnegative measurable function f on Ω such that � ν ( A ) = fdµ A → this permits to define the Radon-Nikodym derivative: → The function f is called the Radon-Nikodym derivative of ν w.r.t. µ , denoted f = dν dµ. • f may not be unique, but if f 0 and f 1 are both Radon-Nikodym derivatives of ν , then f 0 = f 1 almost everywhere µ Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 18 / 70
Definitions Outline Derivatives w.r.t. non-additive measures Derivative (Choquet integral): (non-additive measures) • (Ω , F ) a measurable space, ν, µ : F → R + non-additive measures. → ν is a Choquet integral of µ if there exists a measurable function g : Ω → R + s.t. for all A ∈ F � ν ( A ) = ( C ) gdµ (5) A Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 19 / 70
Definitions Outline Derivatives w.r.t. non-additive measures Derivative (Choquet integral): (non-additive measures) • (Ω , F ) a measurable space, ν, µ : F → R + non-additive measures. → ν is a Choquet integral of µ if there exists a measurable function g : Ω → R + s.t. for all A ∈ F � ν ( A ) = ( C ) gdµ (5) A • µ , ν two non-additive measures. If µ is a Choquet integral of ν , and g is a function such that Equation 5 is satisfied, then dν/dµ = g, → g is a derivative of ν with respect to µ . → Graf and Sugeno studied conditions of when this derivative exists. Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 19 / 70
Definitions Outline Derivatives w.r.t. non-additive measures Derivative (Choquet integral): (Proposition 4 in (Sugeno, 2013)) • Let f ( t ) be a continuous and increasing function with f (0) = 0 , let µ m be a distorted Lebesgue measure, then there exists an increasing � (non-decreasing) function g so that f ( t ) = ( C ) [0 ,t ] g ( τ ) dµ m and the following holds: G ( s ) = F ( s ) /sM ( s ) (6) L − 1 [ F ( s ) /sM ( s )] . g ( t ) = (7) Here, F ( s ) is the Laplace transformation of f , M the Laplace transformation of m , and G the Laplace transformation of g . Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 20 / 70
Definitions Outline Derivatives w.r.t. non-additive measures Computation: • It is possible to compute the Radon-Nikodym derivative (for some examples) • Computations use the Laplace transformation Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 21 / 70
Definitions Outline Derivatives w.r.t. non-additive measures Computation (Example): Applying Proposition 4 (Sugeno, 2013), we have L [ dν p ] = N p ( s ) sM ( s ) = Γ( p + 1) 2 s p − 1 . dµ m Then using the inverse Laplace transform on this expression we obtain: dν p L − 1 [Γ( p + 1) 2 s p − 1 ] = Γ( p + 1) 2Γ( p − 1) t p − 2 = dµ m p ( p − 1) t p − 2 . = 2 Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 22 / 70
Outline f -divergence for non-additive measures LiU 2014 23 / 70
f-divergence Outline f -Divergence Given: P , Q two probabilities a.c. w.r.t. a prob. ν . • f -divergence between P and Q w.r.t. ν � dQ � dP/dν � D f,ν ( P, Q ) = dν f dν dQ/dν Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 24 / 70
f-divergence Outline f -Divergence and distances Examples of f -divergence between P and Q w.r.t. ν � dQ � dP/dν � D f,ν ( P, Q ) = dν f dν dQ/dν Some particular distances • Hellinger distance when f ( x ) = (1 − √ x ) 2 , � � 2 � �� � � � 1 dP dQ � H ( P, Q ) = dν − dν 2 dν Here dP/dν and dQ/dν are the Radon-Nikodym derivatives • Variation distance when f ( x ) = | x − 1 | , � � � δ ( P, Q ) = 1 dP dν − dP � � � dν � � 2 dν � enyi distance, χ 2 -distance • Kullback-Leibler, R´ Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 25 / 70
f-divergence Outline f -Divergence: non-additive measures Definition: • µ 1 , µ 2 two non-additive measures that are Choquet integrals of ν . The f -divergence between µ 1 and µ 2 with respect to ν is defined as � dµ 2 � dµ 1 /dν � D f,ν ( µ 1 , µ 2 ) = ( C ) dν f dν dµ 2 /dν Here dµ 1 /dν and dµ 2 /dν are the derivatives of µ 1 and µ 2 . Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 26 / 70
f-divergence Outline f -Divergence and Hellinger distance: non-additive measures Definition: • µ 1 , µ 2 two non-additive measures that are Choquet integrals of ν . The Hellinger distance between µ 1 and µ 2 with respect to ν is defined as � � 2 � �� � � � 1 dµ 1 dµ 2 � dν − H ν ( µ 1 , µ 2 ) = 2( C ) dν dν Here dµ 1 /dν and dµ 2 /dν are the derivatives of µ 1 and µ 2 . Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 27 / 70
f-divergence Outline f -Divergence and variation distance: non-additive measures Definition: • µ 1 , µ 2 two non-additive measures that are Choquet integrals of ν . The Variation distance between µ 1 and µ 2 with respect to ν is defined as � � � δ ν ( µ 1 , µ 2 ) = 1 dµ 1 dν − dµ 2 � � 2( C ) � dν � � dν � Here dµ 1 /dν and dµ 2 /dν are the derivatives of µ 1 and µ 2 . Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 28 / 70
Outline Properties LiU 2014 29 / 70
Distance Outline Distances: properties Properties: • Proper generalization? Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 30 / 70
Distance Outline Distances: properties Properties: • Proper generalization? • Yes: Let ν , µ 1 , µ 2 be three additive measures such that µ 1 and µ 2 are absolutely continuous with respect to ν . Then, D f,ν ( µ 1 , µ 2 ) is the standard f -divergence. Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 30 / 70
Distance Outline Distances: properties Properties: • Proper generalization? • Yes: Let ν , µ 1 , µ 2 be three additive measures such that µ 1 and µ 2 are absolutely continuous with respect to ν . Then, D f,ν ( µ 1 , µ 2 ) is the standard f -divergence. • Also, H ν ( µ 1 , µ 2 ) and δ ν ( µ 1 , µ 2 ) are the Hellinger distance and the variation distance Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 30 / 70
Distance Outline Distances: properties Properties: • Proper generalization? • Yes: Let ν , µ 1 , µ 2 be three additive measures such that µ 1 and µ 2 are absolutely continuous with respect to ν . Then, D f,ν ( µ 1 , µ 2 ) is the standard f -divergence. • Also, H ν ( µ 1 , µ 2 ) and δ ν ( µ 1 , µ 2 ) are the Hellinger distance and the variation distance • D f,ν ( µ 1 , µ 2 ) with appropriate f (i.e., f ( x ) = (1 − √ x ) 2 and f ( x ) = | x − 1 | ) correspond to Hellinger and variation distance. I.e., Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 30 / 70
Distance Outline Distances: properties Properties: • Proper generalization? • Yes: Let ν , µ 1 , µ 2 be three additive measures such that µ 1 and µ 2 are absolutely continuous with respect to ν . Then, D f,ν ( µ 1 , µ 2 ) is the standard f -divergence. • Also, H ν ( µ 1 , µ 2 ) and δ ν ( µ 1 , µ 2 ) are the Hellinger distance and the variation distance • D f,ν ( µ 1 , µ 2 ) with appropriate f (i.e., f ( x ) = (1 − √ x ) 2 and f ( x ) = | x − 1 | ) correspond to Hellinger and variation distance. I.e., � 1 2 D f,ν ( µ 1 , µ 2 ) = H ν ( µ 1 , µ 2 ) . 1 2 D f,ν ( µ 1 , µ 2 ) = δ ν ( µ 1 , µ 2 ) . Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 30 / 70
Distance Outline Distances: properties Properties: • Distance? Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 31 / 70
Distance Outline Distances: properties Properties: • Distance? ◦ f -divergence is not a distance for additive measures (it is not symmetric, it does not no satisfy triangle inequality) Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 31 / 70
Distance Outline Distances: properties Properties: • Distance? ◦ f -divergence is not a distance for additive measures (it is not symmetric, it does not no satisfy triangle inequality) ◦ Hellinger distance, variation distance are distances. (satisfy positiveness, symmetry, and triangular inequality) Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 31 / 70
Distance Outline Distances: properties Properties: • Distance? ◦ f -divergence is not a distance for additive measures (it is not symmetric, it does not no satisfy triangle inequality) ◦ Hellinger distance, variation distance are distances. (satisfy positiveness, symmetry, and triangular inequality) • So, we only consider distance for Hellinger and variation distance Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 31 / 70
Distance Outline Distances: properties Properties: • Distance? ◦ Positiveness: D f,ν ( µ 1 , µ 2 ) = 0 if µ 1 = µ 2 . So, Hellinger and variation distance are positive Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 32 / 70
Distance Outline Distances: properties Properties: • Distance? ◦ Positiveness: D f,ν ( µ 1 , µ 2 ) = 0 if µ 1 = µ 2 . So, Hellinger and variation distance are positive ◦ Symmetry: Hellinger and variation symmetric by definition Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 32 / 70
Distance Outline Distances: properties Properties: • Distance? ◦ Positiveness: D f,ν ( µ 1 , µ 2 ) = 0 if µ 1 = µ 2 . So, Hellinger and variation distance are positive ◦ Symmetry: Hellinger and variation symmetric by definition ◦ Triangular inequality: ⋆ If ν is submodular, then we have H ν ( µ 1 , µ 2 ) + H ν ( µ 2 , µ 3 ) ≥ H ν ( µ 1 , µ 3 ) . Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 32 / 70
Distance Outline Distances: properties Properties: • Distance? ◦ Positiveness: D f,ν ( µ 1 , µ 2 ) = 0 if µ 1 = µ 2 . So, Hellinger and variation distance are positive ◦ Symmetry: Hellinger and variation symmetric by definition ◦ Triangular inequality: ⋆ If ν is submodular, then we have H ν ( µ 1 , µ 2 ) + H ν ( µ 2 , µ 3 ) ≥ H ν ( µ 1 , µ 3 ) . ⋆ Also, if ν is submodular, then we have δ ν ( µ 1 , µ 2 ) + δ ν ( µ 2 , µ 3 ) ≥ δ ν ( µ 1 , µ 3 ) . Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 32 / 70
Distance Outline Distances: properties Properties: • Triangular inequality. Proof ◦ Proof of triangular inequality for Hellinger distance comes from (seen above) � � � 1 1 1 ( f + g ) 2 dµ ) 2 ≤ (( C ) f 2 d ) g 2 dµ ) 2 + (( C ) (( C ) 2 ◦ Proof of triangular inequality for variation distance comes from (seen above) � � � ( f + g ) dµ ≤ ( C ) ( C ) fdµ + ( C ) gdµ. Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 33 / 70
Distance Outline Distances: properties Properties: • Triangular inequality Hellinger distance. Proof � 2 � 2 � �� � �� � � H ν ( µ 1 , µ 2 ) + H ν ( µ 2 , µ 3 ) = { 1 dµ 1 dµ 2 dν } 1 / 2 + { 1 dµ 2 dµ 3 dν } 1 / 2( C ) dν − 2( C ) dν − dν dν � 2 � 2 � �� �� � � ≥ { 1 dµ 1 dµ 2 dµ 2 dµ 3 dν } 1 / 2 2( C ) dν − + dν − dν dν � 2 � 2 � �� �� � � = { 1 dµ 1 dµ 2 dµ 3 dµ 2 dν } 1 / 2 2( C ) dν − + dν − dν dν � 2 � �� � ≥ { 1 dµ 1 dµ 3 dν } 1 / 2 2( C ) dν − dν = H ν ( µ 1 , µ 3 ) Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 34 / 70
Distance Outline Distances: properties Properties: • Triangular inequality variation distance. Proof δ ν ( µ 1 , µ 2 ) + δ ν ( µ 2 , µ 3 ) = 1 � � dµ 1 dν − dµ 2 � � dν + 1 � � dµ 2 dν − dµ 3 � � � � � 2( C ) 2( C ) � dν � � � � dν dν � � ≥ 1 � �� dµ 1 dν − dµ 2 � � dµ 2 dν − dµ 3 � � � � � � 2( C ) � + dν � � � � dν dν � � � � � � ≥ 1 dµ 1 dν − dµ 3 � � 2( C ) � dν � � dν � = δ ν ( µ 1 , µ 3 ) Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 35 / 70
Distance Outline Distances: properties Properties: • Distance? Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 36 / 70
Distance Outline Distances: properties Properties: • Distance? ◦ If ν is submodular, Hellinger distance is a distance. ◦ If ν is submodular, Variation distance is a distance. Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 36 / 70
Hellinger Distance Outline Hellinger distance: properties Example: Computation of a Hellinger distance between two distorted Lebesgue measures w.r.t. a third one Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 37 / 70
Hellinger Distance Outline Hellinger distance: properties Example: Computation of a Hellinger distance between two distorted Lebesgue measures w.r.t. a third one • Measures: Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 37 / 70
Hellinger Distance Outline Hellinger distance: properties Example: Computation of a Hellinger distance between two distorted Lebesgue measures w.r.t. a third one • Measures: ◦ µ m be the distorted Lebesgue measure with m ( t ) = t 2 , ◦ ν p be the distorted Lebesgue measure with n p ( t ) = t p (i.e., ν p ( A ) = ( λ ( A )) p for p ≥ 2 , and ν p ([0 , t ]) = t p ) Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 37 / 70
Hellinger Distance Outline Hellinger distance: properties Example: Computation of a Hellinger distance between two distorted Lebesgue measures w.r.t. a third one • Measures: ◦ µ m be the distorted Lebesgue measure with m ( t ) = t 2 , ◦ ν p be the distorted Lebesgue measure with n p ( t ) = t p (i.e., ν p ( A ) = ( λ ( A )) p for p ≥ 2 , and ν p ([0 , t ]) = t p ) • Computation: Hellinger distance between ν 2 and ν p w.r.t. µ m . Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 37 / 70
Hellinger Distance Outline Hellinger distance: properties Example: Computation of a Hellinger distance between two distorted Lebesgue measures w.r.t. a third one • Measures: ◦ µ m be the distorted Lebesgue measure with m ( t ) = t 2 , ◦ ν p be the distorted Lebesgue measure with n p ( t ) = t p (i.e., ν p ( A ) = ( λ ( A )) p for p ≥ 2 , and ν p ([0 , t ]) = t p ) • Computation: Hellinger distance between ν 2 and ν p w.r.t. µ m . � 2 � � 1 � � dν 2 dν p � 1 � H µ m ( ν 2 , ν p ) � − = 2( C ) dµ m µ m µ m 0 (8) Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 37 / 70
Hellinger Distance Outline Hellinger distance: properties Example (II): Hellinger distance between ν 2 and ν p w.r.t. µ m where distortions are n p ( t ) = t p and m ( t ) = t 2 . • Recall (from a previous example) that dν p = p ( p − 1) t p − 2 dµ m 2 Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 38 / 70
Hellinger Distance Outline Hellinger distance: properties Example (II): Hellinger distance between ν 2 and ν p w.r.t. µ m where distortions are n p ( t ) = t p and m ( t ) = t 2 . • Recall (from a previous example) that dν p = p ( p − 1) t p − 2 dµ m 2 • Computation (with more Choquet integral – and Laplace transforms): � 2 � � 1 � � dν 2 dν p � 1 � H µ m ( ν 2 , ν p ) � = 2( C ) − dµ m µ m µ m 0 � � 2 � 1 � � � p ( p − 1) � 1 � t ( p − 2) / 2 = 2( C ) 1 − dµ m (9) 2 0 � � 1 − 4 2 p ( p − 1) = (10) ( p + 2) p Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 38 / 70
Hellinger Distance Outline Hellinger distance: properties Example 2: • µ m ′ be the distorted Lebesgue measure with m ′ ( t ) = t . • ν p be the distorted Lebesgue measure with n ( t ) = t p (i.e., ν p ( A ) = ( λ ( A )) p for p ≥ 2 , and ν p ([0 , t ]) = t p ) • Compute the Hellinger distance between ν 2 and ν p w.r.t. µ m ′ . Only difference from Example 1 is µ m ′ Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 39 / 70
Hellinger Distance Outline Hellinger distance: properties Example 2: Hellinger distance between ν 2 and ν p w.r.t. µ m where distortions are n p ( t ) = t p and m ( t ) = t . • First, dν p dµ m ′ = pt p − 1 Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 40 / 70
Hellinger Distance Outline Hellinger distance: properties Example 2: Hellinger distance between ν 2 and ν p w.r.t. µ m where distortions are n p ( t ) = t p and m ( t ) = t . • First, dν p dµ m ′ = pt p − 1 • Computation (with more Choquet integral – and Laplace transforms): � 2 � � 1 � � dν 2 dν p � 1 � H µ m ′ ( ν 2 , ν p ) � = 2( C ) − dµ m µ m µ m 0 � � 1 � √ 1 � 2 � pt p − 1 2 t − = 2( C ) dµ m (11) 0 1 − 2 √ 2 p � = (12) p + 2 Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 40 / 70
Hellinger Distance Outline Hellinger distance: properties Properties: • Compare: � � 2 p ( p − 1) 1 − 4 H µ m ( ν 2 , ν p ) = ( p + 2) p 1 − 2 √ 2 p � H µ m ′ ( ν 2 , ν p ) = (13) p + 2 • The Hellinger distance depends on µ m Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 41 / 70
Hellinger Distance Outline Hellinger distance: properties Properties: • Compare: � � 2 p ( p − 1) 1 − 4 H µ m ( ν 2 , ν p ) = ( p + 2) p 1 − 2 √ 2 p � H µ m ′ ( ν 2 , ν p ) = (13) p + 2 • The Hellinger distance depends on µ m Different for additive measures: H ν ( µ 1 , µ 2 ) is independent of ν . Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 41 / 70
Hellinger Distance Outline Hellinger distance: properties Properties related to the previous example: When p → ∞ , H µ m ( ν 2 , ν p ) = 1 and H µ m ′ ( ν 2 , ν p ) = 1 . Both H µ m ( ν 2 , ν p ) and H µ m ′ ( ν 2 , ν p ) are increasing w.r.t. p > 2 , and the following holds • H µ m ( ν 2 , ν p ) ∈ [0 , 1] for all p ≥ 2 , • H µ m ′ ( ν 2 , ν p ) ∈ [0 , 1] for all p ≥ 2 . Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 42 / 70
Hellinger Distance Outline Hellinger distance: properties Properties: • Conjugate of the measure, same distance ? Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 43 / 70
Hellinger Distance Outline Hellinger distance: properties Properties: • Conjugate of the measure, same distance ? ◦ Recall that conjugate of a measure: µ c ( A ) = 1 − µ m ( X \ A ) Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 43 / 70
Hellinger Distance Outline Hellinger distance: properties Properties: • Conjugate of the measure, same distance ? Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 44 / 70
Hellinger Distance Outline Hellinger distance: properties Properties: • Conjugate of the measure, same distance ? ◦ First question, which conjugate in H ν ( µ 1 , µ 2 ) ? Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 44 / 70
Hellinger Distance Outline Hellinger distance: properties Properties: • Conjugate of the measure, same distance ? ◦ First question, which conjugate in H ν ( µ 1 , µ 2 ) ? ⋆ H ν ( µ 1 , µ 2 ) = (?) H ν ( µ c 1 , µ c 2 ) ⋆ H ν ( µ 1 , µ 2 ) = (?) H ν c ( µ c 1 , µ c 2 ) where µ c ( A ) = 1 − µ ( X \ A ) Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 44 / 70
Hellinger Distance Outline Hellinger distance: properties Properties: • Conjugate of the measure, same distance ? ◦ First question, which conjugate in H ν ( µ 1 , µ 2 ) ? ⋆ H ν ( µ 1 , µ 2 ) = (?) H ν ( µ c 1 , µ c 2 ) ⋆ H ν ( µ 1 , µ 2 ) = (?) H ν c ( µ c 1 , µ c 2 ) where µ c ( A ) = 1 − µ ( X \ A ) • Partial answers: Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 44 / 70
Hellinger Distance Outline Hellinger distance: properties Properties: • Conjugate of the measure, same distance ? ◦ First question, which conjugate in H ν ( µ 1 , µ 2 ) ? ⋆ H ν ( µ 1 , µ 2 ) = (?) H ν ( µ c 1 , µ c 2 ) ⋆ H ν ( µ 1 , µ 2 ) = (?) H ν c ( µ c 1 , µ c 2 ) where µ c ( A ) = 1 − µ ( X \ A ) • Partial answers: ◦ Dual of Distorted Lebesgue is Distorted Lebesgue µ c ( A ) = 1 − µ m ( X \ A ) = 1 − m (1 − x ) Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 44 / 70
Hellinger Distance Outline Hellinger distance: properties Properties: • Conjugate of the measure, same distance ? ◦ First question, which conjugate in H ν ( µ 1 , µ 2 ) ? ⋆ H ν ( µ 1 , µ 2 ) = (?) H ν ( µ c 1 , µ c 2 ) ⋆ H ν ( µ 1 , µ 2 ) = (?) H ν c ( µ c 1 , µ c 2 ) where µ c ( A ) = 1 − µ ( X \ A ) • Partial answers: ◦ Dual of Distorted Lebesgue is Distorted Lebesgue µ c ( A ) = 1 − µ m ( X \ A ) = 1 − m (1 − x ) ◦ If ν is submodular, ν c is supermodular So, H ν ( µ 1 , µ 2 ) is a distance but H ν c ( µ c 1 , µ c 2 ) is not Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 44 / 70
Hellinger Distance Outline Hellinger distance: properties Properties: • Conjugate of the measure, same distance ? ◦ First question, which conjugate in H ν ( µ 1 , µ 2 ) ? ⋆ H ν ( µ 1 , µ 2 ) = (?) H ν ( µ c 1 , µ c 2 ) ⋆ H ν ( µ 1 , µ 2 ) = (?) H ν c ( µ c 1 , µ c 2 ) where µ c ( A ) = 1 − µ ( X \ A ) • Partial answers: ◦ Dual of Distorted Lebesgue is Distorted Lebesgue µ c ( A ) = 1 − µ m ( X \ A ) = 1 − m (1 − x ) ◦ If ν is submodular, ν c is supermodular So, H ν ( µ 1 , µ 2 ) is a distance but H ν c ( µ c 1 , µ c 2 ) is not Therefore, only H ν ( µ 1 , µ 2 ) = (?) H ν ( µ c 1 , µ c 2 ) makes sense Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 44 / 70
Hellinger Distance Outline Hellinger distance: properties Properties: • Conjugate of the measure, same distance ? ◦ First question, which conjugate in H ν ( µ 1 , µ 2 ) ? ⋆ H ν ( µ 1 , µ 2 ) = (?) H ν ( µ c 1 , µ c 2 ) ⋆ H ν ( µ 1 , µ 2 ) = (?) H ν c ( µ c 1 , µ c 2 ) where µ c ( A ) = 1 − µ ( X \ A ) • Partial answers: ◦ Dual of Distorted Lebesgue is Distorted Lebesgue µ c ( A ) = 1 − µ m ( X \ A ) = 1 − m (1 − x ) ◦ If ν is submodular, ν c is supermodular So, H ν ( µ 1 , µ 2 ) is a distance but H ν c ( µ c 1 , µ c 2 ) is not Therefore, only H ν ( µ 1 , µ 2 ) = (?) H ν ( µ c 1 , µ c 2 ) makes sense ◦ This case, difficult (work in progress) E.g., if m ( x ) = x 2 , then m c ( x ) = 2 x − x 2 . Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 44 / 70
Outline Some definitions (II): The Sugeno integral LiU 2014 45 / 70
Definitions Outline Definitions: integrals Sugeno integral (Sugeno, 1974): • µ a non-additive measure, g a measurable function. The Sugeno integral of g w.r.t. µ , where µ g ( r ) := µ ( { x | g ( x ) > r } ) : � [ r ∧ µ g ( r )] . ( S ) gdµ := sup (14) r ∈ [0 , 1] (a) (b) ( ) f ( x ) f ( x ) s ( i ) � ( A ) s ( i ) R ( S ) f d� f ( x ) f ( x ) s ( i ) � ( A ) s ( i ) � ( A ) LiU 2014 46 / 70
Definitions Outline Definitions: integrals Sugeno integral. Discrete version • µ a non-additive measure, f a measurable function. The Sugeno integral of f w.r.t. µ , � ( S ) fdµ = max i =1 ,N min( f ( x s ( i ) ) , µ ( A s ( i ) )) , where f ( x s ( i ) ) indicates that the indices have been permuted so that 0 ≤ f ( x s (1) ) ≤ ... ≤ f ( x s ( N ) ) ≤ 1 and A s ( i ) = { x s ( i ) , ..., x s ( N ) } . LiU 2014 47 / 70
Definitions Outline Definitions: properties Properties: ( X a reference set, a a value in [0 , 1] ) • f, g functions f, g : X → [0 , 1] . Then, ◦ I is minimum homogeneous if and only if, for comonotonic f and g , I ( a ∧ f ) = a ∧ I ( f ) ◦ I is comonotonic maxitive if and only if, for comonotonic f and g , I ( f ∨ g ) = I ( f ) ∨ I ( g ) LiU 2014 48 / 70
Definitions Outline Definitions: properties Characterization of the Sugeno integral • Theorem (Ralescu and Sugeno, 1996; Marichal, 2000; Benvenuti and Mesiar, 2000). Let I : [0 , 1] n → R + be a functional with the following properties ◦ I is comonotonic monotone ◦ I is comonotonic maxitive ◦ I is minimum homogeneous ◦ I (1 , . . . , 1) = 1 Then, there exists a fuzzy measure µ such that I ( f ) is the Sugeno integral of f with respect to µ . LiU 2014 49 / 70
Outline Applications LiU 2014 50 / 70
Applications Outline Aggregation operators Independence. • Choquet integral and Mahalanobis distance ◦ Mahalanobis: covariance matrix ◦ Choquet integral: fuzzy measure • In a single framework: Mahalanobis and Choquet distance Mahalanobis Distance Choquet Integral Covariance Weighted Mean Matrix Fuzzy measure Additive measure Diagonal Matrix LiU 2014 51 / 70
Applications Outline Aggregation operators Independence. • Choquet integral and Mahalanobis distance ◦ Mahalanobis: covariance matrix ◦ Choquet integral: fuzzy measure • A generalization: Choquet-Mahalanobis distance/distribution LiU 2014 52 / 70
Applications Outline Record Linkage Record Linkage: A (protected / public) B (intruder) r 1 s 1 r a a 1 a n s b quasi- a 1 a n i 1 , i 2 , ... confidential identifiers quasi- identifiers identifiers Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 53 / 70
Applications Outline Record Linkage Record Linkage: N � Minimize K i (15) i =1 Subject to : CI µ ( d ( V 1 ( a i ) , V 1 ( b j )) , . . . , d ( V n ( a i ) , V n ( b j ))) − − CI µ ( d ( V 1 ( a i ) , V 1 ( b i )) , . . . , d ( V n ( a i ) , V n ( b i ))) + CK i > 0 ∀ i ∀ j (16) K i ∈ { 0 , 1 } (17) µ ( A ) ∈ [0 , 1] (18) µ ( A ) ≤ µ ( B ) ∀ A, B s.t. A ⊆ B ⊆ X (19) Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 54 / 70
Applications Outline Decision Decision: • Different alternatives • Users have preferences (an order on the alternatives ≺ ) • GOAL: We want to model these preferences (to model ≺ ) Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 55 / 70
Applications Outline Decision under certainty Decision under certainty. Multicriteria decision making • Alternatives expressed in terms of utility functions • Select best alternative by: Step 1. Aggregate utilities: Choquet integral for non-independence Step 2. Rank according to aggregated utilities Criteria Satisfaction on: alt Price Quality Comfort alt Consensus alt Ranking FordT 0.2 0.8 0.3 FordT 0.35 206 0.72 FordT 0.35 206 0.7 0.7 0.8 206 0.72 ... ... ... ... ... ... Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 56 / 70
Applications Outline Decision under uncertainty Decision under uncertainty. • Decision theory based on probability and utility functions to model lack of knowledge (Savage, 1954; Ramsey and von Neumann): ◦ classical/subjective expected utility Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 57 / 70
Applications Outline Decision under uncertainty Decision under uncertainty. • Decision theory based on probability and utility functions to model lack of knowledge (Savage, 1954; Ramsey and von Neumann): ◦ classical/subjective expected utility • Ellsberg paradox: people behave differently than the model!! ◦ Ellsberg paradox violates the postulates of the theory ◦ Alternative model based on non-additive (fuzzy) measures Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 57 / 70
Applications Outline Decision under uncertainty: Ellsberg paradox Decision making: (Ellsberg, 1961) 90 balls in an urn • A player and different games, which prefer? ( f R , f B , ... ) Color of balls Red Black Yellow Number of balls 30 60 f R $ 100 0 0 f B $ 0 $ 100 0 f RY $ 100 0 $ 100 f BY $ 0 $ 100 $ 100 Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 58 / 70
Applications Outline Decision under uncertainty: Ellsberg paradox • How we model ≺ with classical expected utility ? ◦ a (finite) state space S ( options = the balls) ◦ a (finite) set of outcomes X ( benefits = the money) Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 59 / 70
Applications Outline Decision under uncertainty: Ellsberg paradox • How we model ≺ with classical expected utility ? ◦ a (finite) state space S ( options = the balls) ◦ a (finite) set of outcomes X ( benefits = the money) ◦ P be a probability measure on ( X, A , P ) ( P on the balls) ◦ u : X → R + be a utility function (utility of the money) Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 59 / 70
Applications Outline Decision under uncertainty: Ellsberg paradox • How we model ≺ with classical expected utility ? ◦ a (finite) state space S ( options = the balls) ◦ a (finite) set of outcomes X ( benefits = the money) ◦ P be a probability measure on ( X, A , P ) ( P on the balls) ◦ u : X → R + be a utility function (utility of the money) ◦ a function from S to X (an act), F the set of acts (the alternatives). ◦ User preferences on F = { f | f : S → X } denoted by ≺ Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 59 / 70
Applications Outline Decision under uncertainty: Ellsberg paradox • How we model ≺ with classical expected utility ? ◦ a (finite) state space S ( options = the balls) ◦ a (finite) set of outcomes X ( benefits = the money) ◦ P be a probability measure on ( X, A , P ) ( P on the balls) ◦ u : X → R + be a utility function (utility of the money) ◦ a function from S to X (an act), F the set of acts (the alternatives). ◦ User preferences on F = { f | f : S → X } denoted by ≺ ◦ ≺ is represented by P and u when (user preference model) E ( u ( f )) < E ( u ( g )) if and only if f ≺ g where � � u ( x ) P ( f − 1 ( x )) . u ( f ( s )) P ( { s } ) = E ( u ( f )) = s ∈ S x ∈ X Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 59 / 70
Applications Outline Decision under uncertainty: Ellsberg paradox • Computation of the expected utility for a particular act (alternative) ◦ S = { Red, Black, Y ellow } ◦ f RY = ( 0 for a Black, $ 100 for Red, and $ for Yellow ) u (0) P ( f − 1 (0)) + u (100) P ( f − 1 (100)) E ( u ( f RY )) = u (0) P ( { B } ) + u (100) P ( { Y, R } ) = u (0) P ( { B } ) + u (100) P ( { Y } ) + u (100) P ( { R } ) = ◦ Problem. Given a player, and preferences ≺ , determine P and u ◦ E.g., P ( x ) = 1 / 3 and u ( x ) = x . Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 60 / 70
Applications Outline Decision under uncertainty: Ellsberg paradox Decision making: (Ellsberg, 1961) 90 balls in an urn • A player and different games, which prefer? ( f R , f B , ... ) Color of balls Red Black Yellow Number of balls 30 60 f R $ 100 0 0 f B $ 0 $ 100 0 f RY $ 100 0 $ 100 f BY $ 0 $ 100 $ 100 • Most people prefer ◦ f B ≺ f R Vicen¸ c Torra; Non-additive measures and integrals LiU 2014 61 / 70
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