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

Conjunctive forks Basic observations Main result On patterns of conjunctive forks atal 1 , Frantiˇ s and Yori Zwols 2 Vaˇ sek Chv´ sek Mat´ uˇ 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 1 Concordia University, Montreal; 2 Google, London

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

Conjunctive forks Hans Reichenbach Basic observations Definition Main result Main problem H. Reichenbach (1891–1953) Stanford Encyclopedia of Philosophy: “perhaps the greatest empiricist of the 20th century”

Conjunctive forks Hans Reichenbach Basic observations Definition Main result Main problem H. Reichenbach (1891–1953) Stanford Encyclopedia of Philosophy: “perhaps the greatest empiricist of the 20th century” http://plato.stanford.edu/entries/reichenbach/

Conjunctive forks Hans Reichenbach Basic observations Definition Main result 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.”

Conjunctive forks Hans Reichenbach Basic observations Definition Main result 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 Hans Reichenbach Basic observations Definition Main result 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

Conjunctive forks Hans Reichenbach Basic observations Definition Main result Main problem An ordered triple ( A , B , C ) of events in a probability space (Ω , P )

Conjunctive forks Hans Reichenbach Basic observations Definition Main result 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 ) .

Conjunctive forks Hans Reichenbach Basic observations Definition Main result 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)

Conjunctive forks Hans Reichenbach Basic observations Definition Main result 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 A ⊥ ⊥ 1 1 C | 1 1 B and 1 Cov ( 1 1 B ) > 0 and Cov ( 1 1 C ) > 0 1 A , 1 1 B , 1

Conjunctive forks Hans Reichenbach Basic observations Definition Main result 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 A ⊥ ⊥ 1 1 C | 1 1 B and 1 Cov ( 1 1 B ) > 0 and Cov ( 1 1 C ) > 0 1 A , 1 1 B , 1 where Cov ( 1 1 A , 1 1 B ) = P ( A B ) − P ( A ) P ( B )

Conjunctive forks Hans Reichenbach Basic observations Definition Main result 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 A ⊥ ⊥ 1 1 C | 1 1 B and 1 Cov ( 1 1 B ) > 0 and Cov ( 1 1 C ) > 0 1 A , 1 1 B , 1 • • A C ❅ ■ � ✒ where Cov ( 1 1 A , 1 1 B ) = P ( A B ) − P ( A ) P ( B ) ❅ � + + ❅ � • B

Conjunctive forks Hans Reichenbach Basic observations Definition Main result Main problem Notation: ( A , B , C ) P ... the triple of events is a conjunctive fork

Conjunctive forks Hans Reichenbach Basic observations Definition Main result Main problem Notation: ( A , B , C ) P ... the triple of events is a conjunctive fork Construction: having N finite and events A i , i ∈ N , let

Conjunctive forks Hans Reichenbach Basic observations Definition Main result Main problem Notation: ( A , B , C ) P ... the triple of events is a conjunctive fork Construction: having N finite and events A i , i ∈ N , let { ( i , j , k ) ∈ N 3 : ( A i , A j , A k ) P } .

Conjunctive forks Hans Reichenbach Basic observations Definition Main result Main problem Notation: ( A , B , C ) P ... the triple of events is a conjunctive fork Construction: having N finite and events A i , i ∈ N , let { ( i , j , k ) ∈ N 3 : ( A i , A j , A k ) P } . Problem: Given a ternary relation R on a ground set N decide whether it is fork representable , thus ( i , j , k ) ∈ R ⇔ ( A i , A j , A k ) P . for some events A i , i ∈ N .

Conjunctive forks Hans Reichenbach Basic observations Definition Main result Main problem Notation: ( A , B , C ) P ... the triple of events is a conjunctive fork Construction: having N finite and events A i , i ∈ N , let { ( i , j , k ) ∈ N 3 : ( A i , A j , A k ) P } . Problem: Given a ternary relation R on a ground set N decide whether it is fork representable , thus ( i , j , k ) ∈ R ⇔ ( A i , A j , A k ) P . for some events A i , i ∈ N . In algebraic language, solve a system of quadratic equations and inequalities.

Conjunctive forks Signs of correlations Basic observations Necessary conditions Main result Nondegenerate version � A , B � � Cov ( 1 1 A , 1 1 B ) = P ( A B ) − P ( A ) P ( B )

Conjunctive forks Signs of correlations Basic observations Necessary conditions Main result Nondegenerate version � A , B � � Cov ( 1 1 A , 1 1 B ) = P ( A B ) − P ( A ) P ( B ) � σ ij ∈ {− 1 , 0 , 1 } : ij ∈ ( N 2 ) � Let σ = be a pattern of signs indexed by the subsets ij with two elements.

Conjunctive forks Signs of correlations Basic observations Necessary conditions Main result Nondegenerate version � A , B � � Cov ( 1 1 A , 1 1 B ) = P ( A B ) − P ( A ) P ( B ) � σ ij ∈ {− 1 , 0 , 1 } : ij ∈ ( N 2 ) � Let σ = 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 A i , i ∈ N , s.t. ij ∈ ( N σ ij = sgn � A i , A j � , 2 ) .

Conjunctive forks Signs of correlations Basic observations Necessary conditions Main result Nondegenerate version ( A , B , C ) P iff ( C , B , A ) P

Conjunctive forks Signs of correlations Basic observations Necessary conditions Main result Nondegenerate version ( A , B , C ) P iff ( C , B , A ) P Lemma 1: � A , B � 2 � � A , A �� B , B � , tight iff 1 1 A , 1 1 B lin. dependent

Conjunctive forks Signs of correlations Basic observations Necessary conditions Main result Nondegenerate version ( A , B , C ) P iff ( C , B , A ) P Lemma 1: � A , B � 2 � � A , A �� B , 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 , C �� B , B � = � A , B �� B , C � .

Conjunctive forks Signs of correlations Basic observations Necessary conditions Main result Nondegenerate version ( A , B , C ) P iff ( C , B , A ) P Lemma 1: � A , B � 2 � � A , A �� B , 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 , C �� B , B � = � A , B �� B , 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 )