Graphical models
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Graphical Models. BRML Chapter 4. t he zoo of graphical models. Markov networks Belief networks Chain graphs (Belief and M arkov ) Factor graphs =>they are graphical models. two steps of process. Modeling identify the relevant variables make structural assumptions Inference

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Graphical Models

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Graphical models

Graphical Models

BRML Chapter 4


T he zoo of graphical models

the zoo of graphical models

  • Markov networks

  • Belief networks

  • Chain graphs (Belief and Markov )

  • Factor graphs

    =>they are graphical models


Two steps of process

two steps of process

Modeling

identify the relevant variables

make structural assumptions

Inference

answer the interest questions with inference algorithm


Markov networks

Markov Networks

  • Factorization

    the joint probability is described with the product of several potential functions

  • Potential

    a non-negative function of the set of variables

    This is not probability itself.

    (Fig4.1)


What is maximal cliques

What is maximal cliques?

cliques:a fully connected subset of nodes

maximal cliques:cliqueswhich cant’t be extended by including one more adjecent node

We define potential functions on each maximal cliques

>Hammersley-Clifford Theorem


Properties of markov networks

Properties of Markov Networks

  • Definition4.4

    marginalizing,conditioning

  • Definition4.5

    separation

    (Fig4.2)


Graphical models

  • Lattice model

  • Ising model

    ->magnet , image analysis


Chain graphical models

Chain Graphical Models

  • Chain Graphs contain both directed and undirected links.(Fig4.6,4.7)

    1)Remove directed edges

    2)Consider the distributions over maximal clique

    3)Moralize and normalize


Factor graphs fig4 8

Factor Graphs(Fig4.8)

  • Factor graphs are used for inference steps.

  • Square nodes represent factors.

    ex)p(A,B,C)=P(A)P(B)P(C|A,B)

A

B

C


Example of factor graphs

Example of factor graphs

f(A,B,C)=P(A)P(B)P(C|A,B)

A

B

f

Variable node

C

Factor node


Example of factor graph 2

Example of factor graph (2)

f1=P(A),f2=P(B),f3=P(C|A,B)

=> P(A,B,C)=f1*f2*f3

A

B

f1

f2

f3

C


Expressiveness of graphical models

Expressiveness of Graphical Models

  • Directed distributions can be represented as undirected ones.

  • But this transformation can lose some information(conditional independency)

  • the converse questions are same.

  • So,we need I-map

    D-map

    perfect-map(see Definition4.11)


References

references

  • BRML chapter 4

  • PRML chapter 8


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