On Minimum Elementary-triplet Bases for Independence Relations - PowerPoint PPT Presentation

On Minimum Elementary-triplet Bases for Independence Relations Janneke H. Bolt and Linda C. van der Gaag July 2019 1 / 8

Probabilistic independence relations A set of triplets � A , B | C � with A , B , C ⊂ V , where a triplet � A , B | C � captures that Pr( A , B | C ) = Pr( A | C ) · Pr( B | C ) for all possible value combinations of A , B , C . Not any subset of all possible triplets V (3) is a probabilistic independence relation. For example, since � A , B | C � implies � B , A | C � , each probabilistic independence relation will either include none or both of these triplets. 2 / 8

Semi-graphoid axioms G1: if � A , B | C � then � B , A | C � G2: if � A , BD | C � then � A , B | C � G3: if � A , BD | C � then � A , B | CD � G4: if � A , B | CD � and � A , D | C � then � A , BD | C � A semi-graphoid independence relation is a subset of triplets J ⊆ V (3) that satisfies the above properties for all sets A , B , C , D ⊆ V . A semi-graphoid independence relation J can be inferred from a starting set of triplets J by repeatedly applying the semi-graphoid axioms. 3 / 8

Elementary triplets Triplets of the form � A , B | C � . A semi-graphoid relationship is fully captured by its elementary triplets. Semi-graphoid axioms for elementary triplets: E1: if � A , B | C � then � B , A | C � E2: if � A , B | C D � and � A , D | C � then � A , B | C � and � A , D | C B � 4 / 8

Bases for semi-graphoid independence relations Dominant triplets. Any triplet of the independence relation can be derived from one triplet in the basis through axioms G1-G3. Elementary triplets. For example: J dominant basis elementary basis � 1 , 2 | ∅ � � 1 , { 2 , 3 } | ∅ � � 1 , 2 | ∅ � � 1 , 2 | 3 � � 1 , 2 | 3 � � 1 , 3 | ∅ � � 1 , 3 | ∅ � � 1 , 3 | 2 � � 1 , 3 | 2 � � 1 , { 2 , 3 } | ∅ � 5 / 8

Minimum elementary triplet bases Redundant information in an elementary triplet basis. E2: if � A , B | C D � and � A , D | C � then � A , B | C � and � A , D | C B � For example, the elementary triplet basis {� 1 , 2 | ∅ � , � 1 , 2 | 3 � , � 1 , 3 | ∅ � , � 1 , 3 | 2 �} can be reduced to {� 1 , 2 | ∅ � , � 1 , 3 | 2 �} Minimally needed: All A , B -combinations present in the independence relation. All cardinalities of C present in the independence relation. Nb. A minimum elementary triplet basis is not unique. One by one removal of triplets does not necessarily yield a minimum basis. 6 / 8

Minimum bases for singleton starting sets The semi-graphoid closure of the triplet �{ A 1 , . . . , A n } , { B 1 , . . . , B m } | C � is also represented bij the elementary triplets � � � � A i , B j | A \{ A i , . . . A n }∪ B \{ B j , . . . B m }∪ C � � i = 1 , . . . , n , j = 1 , . . . , m For example, the semi-graphoid closure of �{ 1 , 2 } , { 3 , 4 } | ∅ � is represented by the elementary triplets {� 1 , 3 | ∅ � , � 1 , 4 | 3 � , � 2 , 3 | 1 � , � 2 , 4 | { 1 , 3 }�} This implies that the semi-graphoid closure of � A , B | C � can be represented by | A | · | B | elementary triplets. 7 / 8

A few questions How efficient are minimum elementary triplet bases compared to dominant triplet bases? How to compute a minimum basis efficiently? Is one by one removal of triplets a good heuristic? 8 / 8