Markov Networks

1 / 8

# Markov Networks - PowerPoint PPT Presentation

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 ).

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

Representation

Probabilistic

Graphical

Models

Independencies

### MarkovNetworks

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

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)}

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
• Theorem: If P factorizes over H, then H is an I-map for P

A

F

D

B

C

E

Independence  Factorization
• Theorem (Hammersley Clifford):

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

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