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Factor Graphs

Factor Graphs. 2005. 5. 20 Young Ki Baik Computer Vision Lab. Seoul National University. Contents. Introduction Sum product algorithm Computing a single marginal function Computing all marginal function Probabilistic modeling Conclusion. Introduction. What is a Factor Graph?

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Factor Graphs

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  1. Factor Graphs 2005. 5. 20 Young Ki Baik Computer Vision Lab. Seoul National University

  2. Contents • Introduction • Sum product algorithm • Computing a single marginal function • Computing all marginal function • Probabilistic modeling • Conclusion

  3. Introduction • What is a Factor Graph? • A factor graph shows how a function of several variables can be factored into a product of “smaller” function.

  4. Introduction • What is a Factor Graph?

  5. Introduction • Why are factor graphs useful? • Factor graphs simplify problems. • Many efficient algorithms can be applied to factor graphs. • Special Feature • Factor graph represent not only variables or constant, but also functions.

  6. Sum product algorithm • Marginal function • “~{x}” notation • Marginal function Object : Get marginal function g(x) using factor graph

  7. Sum product algorithm • Example (Simple Factor Graph) • Let be a function of four variables.

  8. Computing a single marginal function • Example (Simple Factor Graph) • Marginal function for

  9. Computing a single marginal function • Example (Simple Factor Graph) • Marginal function for

  10. Computing a single marginal function • Example (Simple Factor Graph) • Marginal function for and Bottom-up procedure

  11. Computing all marginal functions • Computing all marginal functions problem • In order to compute all marginal functions , we need to calculate single marginal function as much as n times. • Message passing algorithm • Solution for redundancy problem

  12. Computing all marginal functions • Message passing algorithm • Let denote the message sent from node x to node f in the operation. • Let denote the message sent from node f to node x.

  13. Computing all marginal functions • Message passing algorithm • Variable to local function • Local function to variable

  14. Computing all marginal functions • A Detail Example (Message passing algorithm) • The message may be generated in 4 steps. • ➀ ~ ➃ are step of message passing algorithm • Algorithms start from each leaves. ➃ ➂ ➀ ➁ ➀ ➁ ➂ ➃ ➂ ➃ ➁ ➀

  15. Computing all marginal functions • A Detail Example (Message passing algorithm) • Step 1: Variable to local function ➀ ➀ ➀

  16. Computing all marginal functions • A Detail Example (Message passing algorithm) • Step 2: Local function to variable ➁ ➁ ➁

  17. Computing all marginal functions • A Detail Example (Message passing algorithm) • Step 3: Variable to local function ➂ ➂ ➂

  18. Computing all marginal functions • A Detail Example (Message passing algorithm) • Step 4: Local function to variable ➃ ➃ ➃

  19. Computing all marginal functions • A Detail Example (Message passing algorithm) • Termination ➃ ➂ ➀ ➁ ➀ ➁ ➂ ➃ ➂ ➃ ➁ ➀

  20. Probabilistic Modeling • Markov chain

  21. Probabilistic Modeling • Hidden markov model

  22. Conclusion • More information • The closed factor graph • It can be computed by iterative method. • Conclusion • Factor graph can apply to many efficient algorithms. • Factor graph is only a simplifying tool to solve the problems.

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