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Graphical Models

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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  1. Graphical Models BRML Chapter 4

  2. the zoo of graphical models • Markov networks • Belief networks • Chain graphs (Belief and Markov ) • Factor graphs =>they are graphical models

  3. two steps of process Modeling identify the relevant variables make structural assumptions Inference answer the interest questions with inference algorithm

  4. 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)

  5. 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

  6. Properties of Markov Networks • Definition4.4 marginalizing,conditioning • Definition4.5 separation (Fig4.2)

  7. Lattice model • Ising model ->magnet , image analysis

  8. 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

  9. 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

  10. Example of factor graphs f(A,B,C)=P(A)P(B)P(C|A,B) A B f Variable node C Factor node

  11. 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

  12. 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)

  13. references • BRML chapter 4 • PRML chapter 8

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