Several approaches to conditional probability Mirko Navara Center - PowerPoint PPT Presentation

Several approaches to conditional probability Mirko Navara Center for Machine Perception Department of Cybernetics Faculty of Electrical Engineering Czech Technical University CZ-166 27 Praha, Czech Republic http://cmp.felk.cvut.cz/˜navara

Generalizations of classical probability 2/14 The system of events L need not be a σ -algebra.

Generalizations of classical probability 2/14 The system of events L need not be a σ -algebra. Fuzzy logic : L is certain collection of fuzzy sets (a tribe ) or a σ -complete MV-algebra .

Generalizations of classical probability 2/14 The system of events L need not be a σ -algebra. Fuzzy logic : L is certain collection of fuzzy sets (a tribe ) or a σ -complete MV-algebra . Truth is comparative, not restricted to two values { 0 , 1 } .

Generalizations of classical probability 2/14 The system of events L need not be a σ -algebra. Fuzzy logic : L is certain collection of fuzzy sets (a tribe ) or a σ -complete MV-algebra . Truth is comparative, not restricted to two values { 0 , 1 } . Quantum logic : L is an orthomodular lattice ( OML ) or a more general structure ( orthomodular poset , D-poset = effect algebra ).

Generalizations of classical probability 2/14 The system of events L need not be a σ -algebra. Fuzzy logic : L is certain collection of fuzzy sets (a tribe ) or a σ -complete MV-algebra . Truth is comparative, not restricted to two values { 0 , 1 } . Quantum logic : L is an orthomodular lattice ( OML ) or a more general structure ( orthomodular poset , D-poset = effect algebra ). Two truth values, but limited measurability (if you observe A , you can observe neither B nor its negation ¬ B = non-compatibility of A, B ).

Generalizations of classical probability 2/14 The system of events L need not be a σ -algebra. Fuzzy logic : L is certain collection of fuzzy sets (a tribe ) or a σ -complete MV-algebra . Truth is comparative, not restricted to two values { 0 , 1 } . Quantum logic : L is an orthomodular lattice ( OML ) or a more general structure ( orthomodular poset , D-poset = effect algebra ). Two truth values, but limited measurability (if you observe A , you can observe neither B nor its negation ¬ B = non-compatibility of A, B ). B = the set of classical events of L

Conditional probability of classical events 3/14 Probability measure P ( . ) is considered a mixture of probabilities P ( . | B ) (for the case when B occurs), P ( . |¬ B ) (for the case when B does not occur).

Conditional probability of classical events 3/14 Probability measure P ( . ) is considered a mixture of probabilities P ( . | B ) (for the case when B occurs), P ( . |¬ B ) (for the case when B does not occur). We obtain the formula for total probability P ( A ) = P ( B ) P ( A | B ) + P ( ¬ B ) P ( A |¬ B ) .

Conditional probability of classical events 3/14 Probability measure P ( . ) is considered a mixture of probabilities P ( . | B ) (for the case when B occurs), P ( . |¬ B ) (for the case when B does not occur). We obtain the formula for total probability P ( A ) = P ( B ) P ( A | B ) + P ( ¬ B ) P ( A |¬ B ) . P ( . | B ) , P ( . |¬ B ) are determined by P ( . ) , as unique solutions for A = ( A ∪ B ) ∩ ( A ∪ ¬ B ) = ( A ∩ B ) ∪ ( A ∩ ¬ B ) P ( A ) = P ( A ∩ B ) + P ( A ∩ ¬ B ) = P ( B ) · P ( A | B ) + P ( ¬ B ) · P ( A |¬ B ) P ( A ∩ B ) P ( A | B ) = P ( B )

Conditional probability of classical events 3/14 Probability measure P ( . ) is considered a mixture of probabilities P ( . | B ) (for the case when B occurs), P ( . |¬ B ) (for the case when B does not occur). We obtain the formula for total probability P ( A ) = P ( B ) P ( A | B ) + P ( ¬ B ) P ( A |¬ B ) . P ( . | B ) , P ( . |¬ B ) are determined by P ( . ) , as unique solutions for A = ( A ∪ B ) ∩ ( A ∪ ¬ B ) = ( A ∩ B ) ∪ ( A ∩ ¬ B ) P ( A ) = P ( A ∩ B ) + P ( A ∩ ¬ B ) = P ( B ) · P ( A | B ) + P ( ¬ B ) · P ( A |¬ B ) P ( A ∩ B ) P ( A | B ) = P ( B ) Conditioning in non-classical logic?

Lukasiewicz tribes= σ -complete MV-algebras representable � by fuzzy sets 4/14

Lukasiewicz tribes= σ -complete MV-algebras representable � by fuzzy sets 4/14 admit only integral probability measures � P ( A ) = A d π , where π = P ↾ B is a classical probability measure

Lukasiewicz tribes= σ -complete MV-algebras representable � by fuzzy sets 4/14 admit only integral probability measures � P ( A ) = A d π , where π = P ↾ B is a classical probability measure ⇒ measures depend linearly on membership functions

Lukasiewicz tribes= σ -complete MV-algebras representable � by fuzzy sets 4/14 admit only integral probability measures � P ( A ) = A d π , where π = P ↾ B is a classical probability measure ⇒ measures depend linearly on membership functions ⇒ to satisfy P ( A ) = P ( A ∩ B ) + P ( A ∩ ¬ B ) = P ( B ) · P ( A | B ) + P ( ¬ B ) · P ( A |¬ B ) we need A = ( A ∩ B ) + ( A ∩ ¬ B ) for some intersection ∩

Lukasiewicz tribes= σ -complete MV-algebras representable � by fuzzy sets 5/14 Which combination of operations is good?

Lukasiewicz tribes= σ -complete MV-algebras representable � by fuzzy sets 5/14 Which combination of operations is good? We have ⊙ , ⊕ ... � Lukasiewicz t-norm and t-conorm ∧ , ∨ ... lattice operations = minimum and maximum = G¨ odel (standard) t-norm and t-conorm (all operations applied to fuzzy sets pointwise) A � = ( A ⊙ B ) ⊕ ( A ⊙ ¬ B ) A � = ( A ∧ B ) ∨ ( A ∧ ¬ B ) A � = ( A ⊙ B ) ∨ ( A ⊙ ¬ B ) A � = ( A ∧ B ) ⊕ ( A ∧ ¬ B ) A � = ( A ⊕ B ) ⊙ ( A ⊕ ¬ B ) A � = ( A ∨ B ) ∧ ( A ∨ ¬ B ) · · ·

Lukasiewicz tribes= σ -complete MV-algebras representable � by fuzzy sets 5/14 Which combination of operations is good? We have ⊙ , ⊕ ... � Lukasiewicz t-norm and t-conorm ∧ , ∨ ... lattice operations = minimum and maximum = G¨ odel (standard) t-norm and t-conorm (all operations applied to fuzzy sets pointwise) A � = ( A ⊙ B ) ⊕ ( A ⊙ ¬ B ) A � = ( A ∧ B ) ∨ ( A ∧ ¬ B ) A � = ( A ⊙ B ) ∨ ( A ⊙ ¬ B ) A � = ( A ∧ B ) ⊕ ( A ∧ ¬ B ) A � = ( A ⊕ B ) ⊙ ( A ⊕ ¬ B ) A � = ( A ∨ B ) ∧ ( A ∨ ¬ B ) · · · Why?

Lukasiewicz tribes= σ -complete MV-algebras representable � by fuzzy sets 6/14 There may be an element H s.t. H = ¬ H ( = 1 / 2 ). For B := H , we need the mapping A �→ A ∩ H injective A 0 1 / 2 1 A ∧ H = A ∧ ¬ H 0 1 / 2 1 / 2 A ⊙ H = A ⊙ ¬ H 0 0 1 / 2 A ⊕ H = A ⊕ ¬ H 1 / 2 1 1 A ∨ H = A ∨ ¬ H 1 / 2 1 / 2 1 A + H = A + ¬ H 1 / 2 1 3 / 2 A · H = A · ¬ H 1 / 4 1 / 2 0

Lukasiewicz tribes= σ -complete MV-algebras representable � by fuzzy sets 6/14 There may be an element H s.t. H = ¬ H ( = 1 / 2 ). For B := H , we need the mapping A �→ A ∩ H injective A 0 1 / 2 1 A ∧ H = A ∧ ¬ H 0 1 / 2 1 / 2 A ⊙ H = A ⊙ ¬ H 0 0 1 / 2 A ⊕ H = A ⊕ ¬ H 1 / 2 1 1 A ∨ H = A ∨ ¬ H 1 / 2 1 / 2 1 A + H = A + ¬ H 1 / 2 1 3 / 2 A · H = A · ¬ H 1 / 4 1 / 2 0 Good news: A = ( A · B ) + ( A · ¬ B )

Lukasiewicz tribes= σ -complete MV-algebras representable � by fuzzy sets 6/14 There may be an element H s.t. H = ¬ H ( = 1 / 2 ). For B := H , we need the mapping A �→ A ∩ H injective A 0 1 / 2 1 A ∧ H = A ∧ ¬ H 0 1 / 2 1 / 2 A ⊙ H = A ⊙ ¬ H 0 0 1 / 2 A ⊕ H = A ⊕ ¬ H 1 / 2 1 1 A ∨ H = A ∨ ¬ H 1 / 2 1 / 2 1 A + H = A + ¬ H 1 / 2 1 3 / 2 A · H = A · ¬ H 1 / 4 1 / 2 0 Good news: A = ( A · B ) + ( A · ¬ B ) Even better news: A = ( A · B ) ⊕ ( A · ¬ B ) P ( A ) = P ( A · B ) + P ( A · ¬ B )

σ -complete MV-algebras with product 7/14 [Rieˇ can & Mundici] We need an MV-algebra with product · : L × L → L s.t. (P1) 1 · A = A , (P2) A · ( B ⊖ C ) = ( A · B ) ⊖ ( A · C ) , where A ⊖ B = A ⊙ ¬ B

σ -complete MV-algebras with product 7/14 [Rieˇ can & Mundici] We need an MV-algebra with product · : L × L → L s.t. (P1) 1 · A = A , (P2) A · ( B ⊖ C ) = ( A · B ) ⊖ ( A · C ) , where A ⊖ B = A ⊙ ¬ B If the product exists, it is unique (the same structure is obtained for tribes based on strict Frank t-norms [Klement, Butnariu, Mesiar, MN, H. Weber, Barbieri])

σ -complete MV-algebras with product 8/14 [Rieˇ can, Mundici, Kroupa, Kalina, N´ an´ asiov´ a] We obtain P ( A ) = P ( A · B ) + P ( A · ¬ B ) = P ( B ) · P ( A | B ) + P ( ¬ B ) · P ( A |¬ B ) if we define P ( A | B ) = P ( A · B ) P ( B )

σ -complete MV-algebras with product 8/14 [Rieˇ can, Mundici, Kroupa, Kalina, N´ an´ asiov´ a] We obtain P ( A ) = P ( A · B ) + P ( A · ¬ B ) = P ( B ) · P ( A | B ) + P ( ¬ B ) · P ( A |¬ B ) if we define P ( A | B ) = P ( A · B ) P ( B ) Bad news: P ( B · B ) P ( B | B ) = � = 1 P ( B ) P ( H · H ) P ( H | H ) = = 1 / 2 P ( H )

σ -complete MV-algebras with product 8/14 [Rieˇ can, Mundici, Kroupa, Kalina, N´ an´ asiov´ a] We obtain P ( A ) = P ( A · B ) + P ( A · ¬ B ) = P ( B ) · P ( A | B ) + P ( ¬ B ) · P ( A |¬ B ) if we define P ( A | B ) = P ( A · B ) P ( B ) Bad news: P ( B · B ) P ( B | B ) = � = 1 P ( B ) P ( H · H ) P ( H | H ) = = 1 / 2 P ( H ) “I said neither YES nor NO, but I WAS COMPLETELY RIGHT!”