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Several approaches to conditional probability Mirko Navara Center - - PowerPoint PPT Presentation
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
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Generalizations of classical probability
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.
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Generalizations of classical probability
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}.
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2/14
Generalizations of classical probability
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).
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Generalizations of classical probability
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).
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Generalizations of classical probability
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
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Conditional probability of classical events
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).
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Conditional probability of classical events
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) .
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Conditional probability of classical events
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)
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Conditional probability of classical events
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?
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- Lukasiewicz tribes=σ-complete MV-algebras representable
by fuzzy sets
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- Lukasiewicz tribes=σ-complete MV-algebras representable
by fuzzy sets
admit only integral probability measures P(A) =
- A dπ ,
where π = P ↾B is a classical probability measure
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- Lukasiewicz tribes=σ-complete MV-algebras representable
by fuzzy sets
admit only integral probability measures P(A) =
- A dπ ,
where π = P ↾B is a classical probability measure ⇒ measures depend linearly on membership functions
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- Lukasiewicz tribes=σ-complete MV-algebras representable
by fuzzy sets
admit only integral probability measures P(A) =
- A dπ ,
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 ∩
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- Lukasiewicz tribes=σ-complete MV-algebras representable
by fuzzy sets
Which combination of operations is good?
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- Lukasiewicz tribes=σ-complete MV-algebras representable
by fuzzy sets
Which combination of operations is good? We have ⊙, ⊕ ... Lukasiewicz t-norm and t-conorm ∧, ∨ ... lattice operations = minimum and maximum = G¨
- del (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) · · ·
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- Lukasiewicz tribes=σ-complete MV-algebras representable
by fuzzy sets
Which combination of operations is good? We have ⊙, ⊕ ... Lukasiewicz t-norm and t-conorm ∧, ∨ ... lattice operations = minimum and maximum = G¨
- del (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?
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- Lukasiewicz tribes=σ-complete MV-algebras representable
by fuzzy sets
There may be an element H s.t. H = ¬H (= 1/2). For B := H, we need the mapping A → A ∩ H injective A 1/2 1 A ∧ H = A ∧ ¬H 1/2 1/2 A ⊙ H = A ⊙ ¬H 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
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- Lukasiewicz tribes=σ-complete MV-algebras representable
by fuzzy sets
There may be an element H s.t. H = ¬H (= 1/2). For B := H, we need the mapping A → A ∩ H injective A 1/2 1 A ∧ H = A ∧ ¬H 1/2 1/2 A ⊙ H = A ⊙ ¬H 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 Good news: A = (A · B) + (A · ¬B)
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- Lukasiewicz tribes=σ-complete MV-algebras representable
by fuzzy sets
There may be an element H s.t. H = ¬H (= 1/2). For B := H, we need the mapping A → A ∩ H injective A 1/2 1 A ∧ H = A ∧ ¬H 1/2 1/2 A ⊙ H = A ⊙ ¬H 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 Good news: A = (A · B) + (A · ¬B) Even better news: A = (A · B) ⊕ (A · ¬B) P(A) = P(A · B) + P(A · ¬B)
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σ-complete MV-algebras with product
[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
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σ-complete MV-algebras with product
[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])
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σ-complete MV-algebras with product
[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)
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σ-complete MV-algebras with product
[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) P(B) = 1 P(H|H) = P(H · H) P(H) = 1/2
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σ-complete MV-algebras with product
[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) P(B) = 1 P(H|H) = P(H · H) P(H) = 1/2 “I said neither YES nor NO, but I WAS COMPLETELY RIGHT!”
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σ-complete MV-algebras with product
[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) P(B) = 1 P(H|H) = P(H · H) P(H) = 1/2 “I said neither YES nor NO, but I WAS COMPLETELY RIGHT!” If H has been observed, it gives no information.
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σ-complete MV-algebras with product
[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) P(B) = 1 P(H|H) = P(H · H) P(H) = 1/2 “I said neither YES nor NO, but I WAS COMPLETELY RIGHT!” If H has been observed, it gives no information. ⇒ P(A|H) = P(A)
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Conditioning by classical events of non-classical logics
works because for B ∈ B A = (A ∪ B) ∩ (A ∪ ¬B) A = (A ∩ B) ∪ (A ∩ ¬B) for ∩, ∪ based on any t-norm and t-conorm P(A) = P(A ∩ B) + P(A ∩ ¬B) = P(B) · P(A|B) + P(¬B) · P(A|¬B) where P(A|B) = P(A ∩ B) P(B)
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10/14
Conditionals based on deduction
Inspired by the deduction theorem in fuzzy logic: Theorem: [H´ ajek] Let T be a theory (i.e., a set of formulas) and let ϕ, ψ be formulas. Then T ∪ {ϕ} ⊢ ψ iff there is an n ∈ N such that T ⊢ ϕn → ψ, where ϕn = ϕ ∩ . . . ∩ ϕ
- n times
.
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Conditionals based on deduction
Inspired by the deduction theorem in fuzzy logic: Theorem: [H´ ajek] Let T be a theory (i.e., a set of formulas) and let ϕ, ψ be formulas. Then T ∪ {ϕ} ⊢ ψ iff there is an n ∈ N such that T ⊢ ϕn → ψ, where ϕn = ϕ ∩ . . . ∩ ϕ
- n times
. [Mundici] P(A|B) = P(A ∩ △B) P(△B) where △B = lim
n→∞ Bn
This is conditioning by a classical event △B
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Conditioning in quantum logics
Events are sharp, but not simultaneously observable, A ∩ B need not exist (Given B, we cannot even test A) In orthomodular lattices, A ∧ B exists, but does not have the meaning of conjuction and A = (A ∧ B) ∨ (A ∧ ¬B) A = (A ∨ B) ∧ (A ∨ ¬B) · · · This cannot happen for classical events ⇒ conditioning by classical events works
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Consistent systems of conditional probabilities
[R´ enyi] In classical logic, conditional probabilities P(.|.) : L × Lc → [0, 1] can be the primary notion satisfying axioms, e.g. P
- n
An|B
- =
- n
P(An|B) wheneverAn are mutually disjoint P(B|B) = 1 P(A ∩ C|B) = P(A|B) · P(C|A ∩ B) Non-conditional probability is P(A) = P(A|1) [Colleti, Scozzafava, Chovanec, Drobn´ a, Kˆ
- pka, N´
an´ asiov´ a, Kalina, Khrennikov] Similar systems can be defined in fuzzy and quantum logics. Everything works well, but Problem: Not all non-conditional probabilities can be explained by conditional ones (Conditional probabilities of this kind need not exist even if we have abundance of non-conditional probabilities)
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References
[1] Chovanec, F., Drobn´ a, E., Kˆ
- pka, F., N´
an´ asiov´ a, O.: Conditional states and independence in D-posets. Soft Computing, to appear. DOI 10.1007/s00500-009-0487-0 [2] Flaminio, T., Montagna, F.: A logical and algebraic treatment of conditional probability.
- Arch. Math. Logic 44 (2005), 245–262.
[3] Kalina, M., N´ an´ asiov´ a, O.: Conditional measures and joint distributions on MV-albebras.
- Proc. 10th Int. Conf. Information Processing and Management of Uncertainty,
Perugia, Italy, 2004. [4] Kroupa, T.: Many-dimensional observables on Lukasiewicz tribe: Constructions, conditioning and conditional independence. Kybernetika 41 (2005), 451–468. [5] Kroupa, T.: Conditional probability on MV-algebras. Fuzzy Sets, Syst. 149 (2005), 369–381. [6] Mundici, D.: Faithful and invariant conditional probability in Lukasiewicz logic. Preprint. [7] N´ an´ asiov´ a, O.: Principle Conditioning. Internat. J. Theor. Physics 43 (2004), 1757–1767.
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