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Representation. Probabilistic Graphical Models. Independencies. Markov Networks. Independence & Factorization. Factorization of a distribution P over a graph implies independencies in P. P(X,Y,Z) = P(X) P(Y). X,Y independent. P( X , Y , Z )   1 ( X , Y )  2 ( Y , Z ).

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Presentation Transcript
markov networks

Representation

Probabilistic

Graphical

Models

Independencies

MarkovNetworks

independence factorization
Independence & Factorization
  • Factorization of a distribution P over a graph implies independencies in P

P(X,Y,Z) = P(X) P(Y)

X,Y independent

  • P(X,Y,Z)  1(X,Y) 2(Y,Z)

(X  Y | Z)

separation in mns
Separation in MNs

Definition:

X and Y are separated in H given Z

if there is no active trail in H

between X and Y given Z

I(H) = {(X  Y | Z) : sepH(X, Y | Z)}

slide4

A

F

D

B

C

E

  • If P satisfies I(H), we say that H is an I-map (independency map) of P
factorization independence
Factorization  Independence
  • Theorem: If P factorizes over H, then H is an I-map for P

A

F

D

B

C

E

independence factorization1
Independence  Factorization
  • Theorem (Hammersley Clifford):

For a positive distribution P, if H is an I-map for P, then P factorizes over H

summary
Summary

Two equivalent* views of graph structure:

  • Factorization: H allows P to be represented
  • I-map: Independencies encoded by H hold in P

If P factorizes over a graph H, we can read from the graph

independencies that must hold in P (an independency map)

* for positive distributions