factorization & independence
DESCRIPTION
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. - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Factorization & Independence](https://reader035.vdocuments.mx/reader035/viewer/2022062315/56814e91550346895dbc3a4e/html5/thumbnails/1.jpg)
Daphne Koller
Bayesian Networks
Factorization & Independence
ProbabilisticGraphicalModels
Representation
![Page 2: Factorization & Independence](https://reader035.vdocuments.mx/reader035/viewer/2022062315/56814e91550346895dbc3a4e/html5/thumbnails/2.jpg)
Daphne Koller
Dual View
![Page 3: Factorization & Independence](https://reader035.vdocuments.mx/reader035/viewer/2022062315/56814e91550346895dbc3a4e/html5/thumbnails/3.jpg)
Daphne Koller
Independence Assumptions in G
• The independencies implied by G I(G) =
![Page 4: Factorization & Independence](https://reader035.vdocuments.mx/reader035/viewer/2022062315/56814e91550346895dbc3a4e/html5/thumbnails/4.jpg)
Daphne Koller
G and PWe say that G is an I-map (independence map) of P if
![Page 5: Factorization & Independence](https://reader035.vdocuments.mx/reader035/viewer/2022062315/56814e91550346895dbc3a4e/html5/thumbnails/5.jpg)
Daphne Koller
I-maps
I D Prob
i0 d0 0.42
i0 d1 0.18
i1 d0 0.28
i1 d1 0.12
I D Prob.i0 d0 0.282i0 d1 0.02 i1 d0 0.564i1 d1 0.134
P1P2
![Page 6: Factorization & Independence](https://reader035.vdocuments.mx/reader035/viewer/2022062315/56814e91550346895dbc3a4e/html5/thumbnails/6.jpg)
Daphne Koller
Factorization Independence
Theorem: If P factorizes over G then G is an I-map for P
ID
G
L
S
![Page 7: Factorization & Independence](https://reader035.vdocuments.mx/reader035/viewer/2022062315/56814e91550346895dbc3a4e/html5/thumbnails/7.jpg)
Daphne Koller
P(D,I,G,S,L) = P(D) P(I) P(G | I,D) P(L | G) P(S | I)
![Page 8: Factorization & Independence](https://reader035.vdocuments.mx/reader035/viewer/2022062315/56814e91550346895dbc3a4e/html5/thumbnails/8.jpg)
Daphne Koller
Independence Factorization
Theorem: If G is an I-map for P then P factorizes over G ID
G
L
S
![Page 9: Factorization & Independence](https://reader035.vdocuments.mx/reader035/viewer/2022062315/56814e91550346895dbc3a4e/html5/thumbnails/9.jpg)
Daphne Koller
ID
G
L
S
![Page 10: Factorization & Independence](https://reader035.vdocuments.mx/reader035/viewer/2022062315/56814e91550346895dbc3a4e/html5/thumbnails/10.jpg)
Daphne Koller
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
![Page 11: Factorization & Independence](https://reader035.vdocuments.mx/reader035/viewer/2022062315/56814e91550346895dbc3a4e/html5/thumbnails/11.jpg)
Daphne Koller
END END END