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Factorization & Independence

Representation. Probabilistic Graphical Models. Bayesian Networks. Factorization & Independence. Dual View. Independence Assumptions in G. The independencies implied by G I(G) =. G and P. We say that G is an I-map (independence map) of P if. I-maps. P 2. P 1.

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Factorization & Independence

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  1. Representation Probabilistic Graphical Models Bayesian Networks Factorization & Independence

  2. Dual View

  3. Independence Assumptions in G • The independencies implied by G I(G) =

  4. G and P We say that G is an I-map (independence map) of P if

  5. I-maps P2 P1

  6. Factorization  Independence Theorem: If P factorizes over G then G is an I-map for P D I G S L

  7. P(D,I,G,S,L) = P(D) P(I) P(G | I,D) P(L | G) P(S | I)

  8. Independence  Factorization Theorem: If G is an I-map for P then P factorizes over G D I G S L

  9. D I G S L

  10. Summary • d-separation allows us to use G to read off independencies that must hold in any distribution P that factorizes over G • If the d-separation independencies hold in P, it must be representable as a BN over G

  11. END END END

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