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

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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.

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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.

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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 | CD and A, D | C then A, B | C and A, D | CB

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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} | ∅

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Minimum elementary triplet bases

Redundant information in an elementary triplet basis. E2: if A, B | CD and A, D | C then A, B | C and A, D | CB 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.

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Minimum bases for singleton starting sets

The semi-graphoid closure of the triplet {A1, . . . , An}, {B1, . . . , Bm} | C is also represented bij the elementary triplets

• Ai, Bj | A\{Ai, . . . An}∪ B\{Bj, . . . Bm}∪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.

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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?

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