On patterns of conjunctive forks atal 1 , Franti s and Yori Zwols 2 - - PowerPoint PPT Presentation

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On patterns of conjunctive forks atal 1 , Franti s and Yori Zwols 2 - - PowerPoint PPT Presentation

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Conjunctive forks Basic observations Main result On patterns of conjunctive forks atal 1 , Franti s and Yori Zwols 2 Va sek Chv sek Mat u Institute of Information Theory and Automation Academy of Sciences of the Czech Republic

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Conjunctive forks Basic observations Main result

On patterns of conjunctive forks

Vaˇ sek Chv´ atal1, Frantiˇ sek Mat´ uˇ s and Yori Zwols2

Institute of Information Theory and Automation Academy of Sciences of the Czech Republic matus@utia.cas.cz

Algebraic Statistics June 8-11, 2015 Genova, Italy

1Concordia University, Montreal; 2Google, London

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

  • H. Reichenbach (1891–1953)
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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

  • H. Reichenbach (1891–1953)

Stanford Encyclopedia of Philosophy: “perhaps the greatest empiricist of the 20th century”

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

  • H. Reichenbach (1891–1953)

Stanford Encyclopedia of Philosophy: “perhaps the greatest empiricist of the 20th century”

http://plato.stanford.edu/entries/reichenbach/

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

  • H. Reichenbach (1891–1953)

Stanford Encyclopedia of Philosophy: “perhaps the greatest empiricist of the 20th century”

http://plato.stanford.edu/entries/reichenbach/

Principle of the Common Cause: “If an improbable coincidence has occurred, there must exist a common cause.”

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

  • H. Reichenbach (1891–1953)

Stanford Encyclopedia of Philosophy: “perhaps the greatest empiricist of the 20th century”

http://plato.stanford.edu/entries/reichenbach/

Principle of the Common Cause: “If an improbable coincidence has occurred, there must exist a common cause.” The Direction of Time (1956) University of California Press.

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

  • H. Reichenbach (1891–1953)

Stanford Encyclopedia of Philosophy: “perhaps the greatest empiricist of the 20th century”

http://plato.stanford.edu/entries/reichenbach/

Principle of the Common Cause: “If an improbable coincidence has occurred, there must exist a common cause.” The Direction of Time (1956) University of California Press. conjunctive forks play a central role in Reichenbach’s causal theory of time

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

An ordered triple (A, B, C) of events in a probability space (Ω, P)

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

An ordered triple (A, B, C) of events in a probability space (Ω, P) is a conjunctive fork if P(A ∩ C|B) = P(A|B) · P(C|B) , P(A ∩ C|Ω \ B) = P(A| Ω \ B) · P(C| Ω \ B) , P(A|B) > P(A|Ω \ B) , P(C|B) > P(C|Ω \ B) .

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

An ordered triple (A, B, C) of events in a probability space (Ω, P) is a conjunctive fork if P(A ∩ C|B) = P(A|B) · P(C|B) , P(A ∩ C|Ω \ B) = P(A| Ω \ B) · P(C| Ω \ B) , P(A|B) > P(A|Ω \ B) , P(C|B) > P(C|Ω \ B) . (implicit assumption 0 < P(B) < 1)

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

An ordered triple (A, B, C) of events in a probability space (Ω, P) is a conjunctive fork if P(A ∩ C|B) = P(A|B) · P(C|B) , P(A ∩ C|Ω \ B) = P(A| Ω \ B) · P(C| Ω \ B) , P(A|B) > P(A|Ω \ B) , P(C|B) > P(C|Ω \ B) . (implicit assumption 0 < P(B) < 1) In contemporary language, 1 1 A⊥ ⊥1 1 C|1 1 B and Cov(1 1 A, 1 1 B) > 0 and Cov(1 1 B, 1 1 C) > 0

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

An ordered triple (A, B, C) of events in a probability space (Ω, P) is a conjunctive fork if P(A ∩ C|B) = P(A|B) · P(C|B) , P(A ∩ C|Ω \ B) = P(A| Ω \ B) · P(C| Ω \ B) , P(A|B) > P(A|Ω \ B) , P(C|B) > P(C|Ω \ B) . (implicit assumption 0 < P(B) < 1) In contemporary language, 1 1 A⊥ ⊥1 1 C|1 1 B and Cov(1 1 A, 1 1 B) > 0 and Cov(1 1 B, 1 1 C) > 0 where Cov(1 1 A, 1 1 B) = P(A B) − P(A)P(B)

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

An ordered triple (A, B, C) of events in a probability space (Ω, P) is a conjunctive fork if P(A ∩ C|B) = P(A|B) · P(C|B) , P(A ∩ C|Ω \ B) = P(A| Ω \ B) · P(C| Ω \ B) , P(A|B) > P(A|Ω \ B) , P(C|B) > P(C|Ω \ B) . (implicit assumption 0 < P(B) < 1) In contemporary language, 1 1 A⊥ ⊥1 1 C|1 1 B and Cov(1 1 A, 1 1 B) > 0 and Cov(1 1 B, 1 1 C) > 0 where Cov(1 1 A, 1 1 B) = P(A B) − P(A)P(B)

❅ ❅ ❅ ■

  • ✒

A C B

  • +

+

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

Notation: (A, B, C)P ... the triple of events is a conjunctive fork

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

Notation: (A, B, C)P ... the triple of events is a conjunctive fork Construction: having N finite and events Ai, i ∈ N, let

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

Notation: (A, B, C)P ... the triple of events is a conjunctive fork Construction: having N finite and events Ai, i ∈ N, let {(i, j, k) ∈ N3 : (Ai, Aj, Ak)P} .

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

Notation: (A, B, C)P ... the triple of events is a conjunctive fork Construction: having N finite and events Ai, i ∈ N, let {(i, j, k) ∈ N3 : (Ai, Aj, Ak)P} . Problem: Given a ternary relation R on a ground set N decide whether it is fork representable, thus (i, j, k) ∈ R ⇔ (Ai, Aj, Ak)P . for some events Ai, i ∈ N.

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Conjunctive forks Basic observations Main result Hans Reichenbach Definition Main problem

Notation: (A, B, C)P ... the triple of events is a conjunctive fork Construction: having N finite and events Ai, i ∈ N, let {(i, j, k) ∈ N3 : (Ai, Aj, Ak)P} . Problem: Given a ternary relation R on a ground set N decide whether it is fork representable, thus (i, j, k) ∈ R ⇔ (Ai, Aj, Ak)P . for some events Ai, i ∈ N. In algebraic language, solve a system of quadratic equations and inequalities.

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

A, B Cov(1 1 A, 1 1 B) = P(A B) − P(A)P(B)

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

A, B Cov(1 1 A, 1 1 B) = P(A B) − P(A)P(B) Let σ =

  • σij ∈ {−1, 0, 1}: ij ∈ (N

2)

be a pattern of signs indexed by the subsets ij with two elements.

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

A, B Cov(1 1 A, 1 1 B) = P(A B) − P(A)P(B) Let σ =

  • σij ∈ {−1, 0, 1}: ij ∈ (N

2)

be a pattern of signs indexed by the subsets ij with two elements. A simpler problem is easily solvable: given any pattern σ, there exist events Ai, i ∈ N, s.t. σij = sgnAi, Aj , ij ∈ (N

2) .

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

(A, B, C)P iff (C, B, A)P

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

(A, B, C)P iff (C, B, A)P Lemma 1: A, B2 A, AB, B, tight iff 1 1 A, 1 1 B lin. dependent

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

(A, B, C)P iff (C, B, A)P Lemma 1: A, B2 A, AB, B, tight iff 1 1 A, 1 1 B lin. dependent Lemma 2: If 1 1 A⊥ ⊥1 1 C|1 1 B then A, CB, B = A, BB, C.

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

(A, B, C)P iff (C, B, A)P Lemma 1: A, B2 A, AB, B, tight iff 1 1 A, 1 1 B lin. dependent Lemma 2: If 1 1 A⊥ ⊥1 1 C|1 1 B then A, CB, B = A, BB, C. Corollary 1: (A, B, C)P implies that A, B, C are nontrivial ( thus (A, A, A)P, (B, B, B)P and (C, C, C)P ) and any two are positively correlated ( thus (A, B, B)P, (B, C, C)P, (C, A, A)P )

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

(A, B, C)P iff (C, B, A)P Lemma 1: A, B2 A, AB, B, tight iff 1 1 A, 1 1 B lin. dependent Lemma 2: If 1 1 A⊥ ⊥1 1 C|1 1 B then A, CB, B = A, BB, C. Corollary 1: (A, B, C)P implies that A, B, C are nontrivial ( thus (A, A, A)P, (B, B, B)P and (C, C, C)P ) and any two are positively correlated ( thus (A, B, B)P, (B, C, C)P, (C, A, A)P ) Corollary 2: (A, B, C)P and (A, C, B)P implies B = C

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

(A, B, C)P iff (C, B, A)P Lemma 1: A, B2 A, AB, B, tight iff 1 1 A, 1 1 B lin. dependent Lemma 2: If 1 1 A⊥ ⊥1 1 C|1 1 B then A, CB, B = A, BB, C. Corollary 1: (A, B, C)P implies that A, B, C are nontrivial ( thus (A, A, A)P, (B, B, B)P and (C, C, C)P ) and any two are positively correlated ( thus (A, B, B)P, (B, C, C)P, (C, A, A)P ) Corollary 2: (A, B, C)P and (A, C, B)P implies B = C Proof: A, CB, B = A, BB, C and A, BC, C = A, CB, C combine to B, C2 = B, BC, C, then B = C

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

To be fork representable, R ⊆ N3 must satisfy the symmetry (i, j, k) ∈ R ⇒ (k, j, i) ∈ R

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

To be fork representable, R ⊆ N3 must satisfy the symmetry (i, j, k) ∈ R ⇒ (k, j, i) ∈ R by Corollary 2 (i, j, k) ∈ R and (i, k, j) ∈ R ⇒ j = k

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

To be fork representable, R ⊆ N3 must satisfy the symmetry (i, j, k) ∈ R ⇒ (k, j, i) ∈ R by Corollary 2 (i, j, k) ∈ R and (i, k, j) ∈ R ⇒ j = k symmetry and ⇔ ... betweenness

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

To be fork representable, R ⊆ N3 must satisfy the symmetry (i, j, k) ∈ R ⇒ (k, j, i) ∈ R by Corollary 2 (i, j, k) ∈ R and (i, k, j) ∈ R ⇒ j = k symmetry and ⇔ ... betweenness by Corollary 1 (i, j, k) ∈ R ⇒ (i, i, i) ∈ R , (j, j, j) ∈ R , (k, k, k) ∈ R , (i, j, k) ∈ R ⇒ (i, j, j) ∈ R , (j, k, k) ∈ R and (k, i, i) ∈ R

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

To be fork representable, R ⊆ N3 must satisfy the symmetry (i, j, k) ∈ R ⇒ (k, j, i) ∈ R by Corollary 2 (i, j, k) ∈ R and (i, k, j) ∈ R ⇒ j = k symmetry and ⇔ ... betweenness by Corollary 1 (i, j, k) ∈ R ⇒ (i, i, i) ∈ R , (j, j, j) ∈ R , (k, k, k) ∈ R , (i, j, k) ∈ R ⇒ (i, j, j) ∈ R , (j, k, k) ∈ R and (k, i, i) ∈ R Collecting the four implications ... weak betweenness

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

Lemma 3: (A, B, A)P iff A = B is nontrivial

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

Lemma 3: (A, B, A)P iff A = B is nontrivial In the main problem assume R ⊆ N3 satisfies

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

Lemma 3: (A, B, A)P iff A = B is nontrivial In the main problem assume R ⊆ N3 satisfies (i, j, i) ∈ R iff i = j for i ∈ N

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

Lemma 3: (A, B, A)P iff A = B is nontrivial In the main problem assume R ⊆ N3 satisfies (i, j, i) ∈ R iff i = j for i ∈ N which excludes the trivial events and cloning ... the assumption of nondegeneracy

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

Lemma 3: (A, B, A)P iff A = B is nontrivial In the main problem assume R ⊆ N3 satisfies (i, j, i) ∈ R iff i = j for i ∈ N which excludes the trivial events and cloning ... the assumption of nondegeneracy In a weak betweenness R for any 3-set ijk at most one of (i, j, k), (j, k, i), (k, i, j) belongs to R

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Conjunctive forks Basic observations Main result Signs of correlations Necessary conditions Nondegenerate version

Lemma 3: (A, B, A)P iff A = B is nontrivial In the main problem assume R ⊆ N3 satisfies (i, j, i) ∈ R iff i = j for i ∈ N which excludes the trivial events and cloning ... the assumption of nondegeneracy In a weak betweenness R for any 3-set ijk at most one of (i, j, k), (j, k, i), (k, i, j) belongs to R R is called solvable if and only if the system xik = xij + xjk for (i, j, k) ∈ R pairwise distinct, has a solution with all involved xij positive.

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Conjunctive forks Basic observations Main result Theorem Solvability Open problems

Theorem Under nondegeneracy, R ⊆ N3 is fork representable iff it is a solvable weak betweenness.

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Conjunctive forks Basic observations Main result Theorem Solvability Open problems

Theorem Under nondegeneracy, R ⊆ N3 is fork representable iff it is a solvable weak betweenness. In general, R must be a ’regular’ weak betweenness. Then a quotient Q of R is constructed. It is a weak betweenness that satisfies the nondegeneracy

  • condition. R is fork representable iff Q is solvable.
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Conjunctive forks Basic observations Main result Theorem Solvability Open problems

Theorem Under nondegeneracy, R ⊆ N3 is fork representable iff it is a solvable weak betweenness. In general, R must be a ’regular’ weak betweenness. Then a quotient Q of R is constructed. It is a weak betweenness that satisfies the nondegeneracy

  • condition. R is fork representable iff Q is solvable.

The conditions can be verified in time polynomial in |N|.

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Conjunctive forks Basic observations Main result Theorem Solvability Open problems

necessity

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Conjunctive forks Basic observations Main result Theorem Solvability Open problems

necessity Assume nondegeneracy and R to be fork representable by Ai, i ∈ N.

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Conjunctive forks Basic observations Main result Theorem Solvability Open problems

necessity Assume nondegeneracy and R to be fork representable by Ai, i ∈ N. For i, k different, participating in a fork, let xik = ln Ai,Ai1/2 Ak,Ak1/2

Ai,Ak

.

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Conjunctive forks Basic observations Main result Theorem Solvability Open problems

necessity Assume nondegeneracy and R to be fork representable by Ai, i ∈ N. For i, k different, participating in a fork, let xik = ln Ai,Ai1/2 Ak,Ak1/2

Ai,Ak

. By Lemma 1, xik > 0. By Lemma 2, R is solvable since xik = ln Ai,Ai1/2 Aj,Aj1/2

Ai,Aj Aj,Aj1/2 Ak,Ak1/2 Aj,Ak

= xij + xjk

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Conjunctive forks Basic observations Main result Theorem Solvability Open problems

necessity Assume nondegeneracy and R to be fork representable by Ai, i ∈ N. For i, k different, participating in a fork, let xik = ln Ai,Ai1/2 Ak,Ak1/2

Ai,Ak

. By Lemma 1, xik > 0. By Lemma 2, R is solvable since xik = ln Ai,Ai1/2 Aj,Aj1/2

Ai,Aj Aj,Aj1/2 Ak,Ak1/2 Aj,Ak

= xij + xjk for (Ai, Aj, Ak)P.

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Conjunctive forks Basic observations Main result Theorem Solvability Open problems

necessity Assume nondegeneracy and R to be fork representable by Ai, i ∈ N. For i, k different, participating in a fork, let xik = ln Ai,Ai1/2 Ak,Ak1/2

Ai,Ak

. By Lemma 1, xik > 0. By Lemma 2, R is solvable since xik = ln Ai,Ai1/2 Aj,Aj1/2

Ai,Aj Aj,Aj1/2 Ak,Ak1/2 Aj,Ak

= xij + xjk for (Ai, Aj, Ak)P. sufficiency

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Conjunctive forks Basic observations Main result Theorem Solvability Open problems

necessity Assume nondegeneracy and R to be fork representable by Ai, i ∈ N. For i, k different, participating in a fork, let xik = ln Ai,Ai1/2 Ak,Ak1/2

Ai,Ak

. By Lemma 1, xik > 0. By Lemma 2, R is solvable since xik = ln Ai,Ai1/2 Aj,Aj1/2

Ai,Aj Aj,Aj1/2 Ak,Ak1/2 Aj,Ak

= xij + xjk for (Ai, Aj, Ak)P. sufficiency P is constructed on ZN

2 explicitly, can be arbitrarily close to the

uniform distribution; Fourier-Stieltjes transform of P is related to solvability + few other tricks

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Conjunctive forks Basic observations Main result Theorem Solvability Open problems

Given a pattern of signs

  • σij : ij ∈ (N

2)

and a family { (i, j, k)}, represent them simultaneously by events.

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Conjunctive forks Basic observations Main result Theorem Solvability Open problems

Given a pattern of signs

  • σij : ij ∈ (N

2)

and a family { (i, j, k)}, represent them simultaneously by events. For P > 0 likely in reach (including primary decomposition), for P 0 open ??

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Conjunctive forks Basic observations Main result Theorem Solvability Open problems

Given a pattern of signs

  • σij : ij ∈ (N

2)

and a family { (i, j, k)}, represent them simultaneously by events. For P > 0 likely in reach (including primary decomposition), for P 0 open ?? Given a family { (i, j, k)}, represent it by arbitrary variables (P > 0 or P 0) ??

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Conjunctive forks Basic observations Main result Theorem Solvability Open problems

Given a pattern of signs

  • σij : ij ∈ (N

2)

and a family { (i, j, k)}, represent them simultaneously by events. For P > 0 likely in reach (including primary decomposition), for P 0 open ?? Given a family { (i, j, k)}, represent it by arbitrary variables (P > 0 or P 0) ?? Gaussian case is likely not difficult.