Independence in Markov Networks
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Transcript of Independence in Markov Networks
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Daphne Koller
Markov Networks
Independencein Markov Networks
ProbabilisticGraphicalModels
Representation
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Daphne Koller
Influence Flow in Undirected Graph
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Daphne Koller
Separation in Undirected Graph
• A trail X1—X2—… —Xk-1—Xk is active given Z
• X and Y are separated in H given Z if
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Daphne Koller
Independences in Undirected Graph
• The independences implied by H
I(H) =
• We say that H is an I-map (independence map) of P if
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Daphne Koller
FactorizationP factorizes over H
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Daphne Koller
Factorization Independence
Theorem: If P factorizes over H then H is an I-map for P
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Daphne Koller
BD
C
A
E
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Daphne Koller
Independence Factorization
Theorem: If H is an I-map for P then P factorizes over H
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Daphne Koller
Independence Factorization
Hammersley-Clifford Theorem: If H is an I-map for P, and P is positive, then P factorizes over H
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Daphne Koller
Summary• Separation in Markov network H allows us to
“read off” independence properties that hold in any Gibbs distribution that factorizes over H
• Although the same graph can correspond to different factorizations, they have the same independence properties