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# Factor Graph & Sum-Product Algorithm

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### Factor Graph & Sum-Product Algorithm

1. 1. Factor Graph , Sum-Product Algorithm A Mini Demo TonyOuyang, ACM Honored Class 2009 Spring
2. 2. <ul><li>Graphical Model </li></ul><ul><li>Factor Graph </li></ul>02
3. 3. <ul><li>Graphical Model </li></ul><ul><ul><li>Graph Theory + Probability </li></ul></ul><ul><li>Factor Graph </li></ul><ul><ul><li>a function of several variables can be factored into a product of &quot;smaller&quot; functions. </li></ul></ul>g(x,y)=xy+x g(x,y)=f1(x)f2(y)‏ f1(x)=x f2(y)=y+1.
4. 4. <ul><li>Very general </li></ul><ul><li>Simplify problem </li></ul><ul><li>Many efficient algorithms </li></ul>Why are factor graph useful ? <ul><ul><li>Demo (with a “sum-product-algorithm heart”)‏ </li></ul></ul><ul><ul><li>@ http://www.endora.ch/factor.html </li></ul></ul>
5. 5. <ul><li>Message Passing Algorithm </li></ul><ul><li>Marginal Functions </li></ul><ul><li>Sum-Product Algorithm </li></ul><ul><li>Constraints & Indicator Functions </li></ul>A closer look
6. 6. <ul><li>e.g. Counting Problem </li></ul><ul><ul><li>Naïve mode : large amount of msgs, long distance ... </li></ul></ul><ul><ul><li>Msg passing mode : simple info handling, only care about adjacent people </li></ul></ul>Msg Passing Algorithm Reference : Information Theory, Inference, and_Learning Algorithms, chp16
7. 7. <ul><li>More complicated case ... </li></ul>Msg Passing Algorithm con't
8. 8. <ul><li>Path counting , a familiar case </li></ul>Msg Passing Algorithm con't # How many such paths are there from A to B? # If a random path from A to B is selected, what is the probability that it passes through a particular node in the grid?
9. 9. Reference : Bishop, chp8 Why efficient ? # global -> local # interchange summation & production # reuse ! * fact : dynamic programming
10. 10. <ul><li>Back to example </li></ul><ul><ul><li>Constraints and Indicator Functions </li></ul></ul><ul><ul><li>What's the factor ? (against random variable)‏ </li></ul></ul>&quot;no pressure&quot; = &quot;pump switched off&quot; OR &quot;pump defect&quot;. define the function f1 : f1(np,pso,pd) = 1 if the constraint is satisfied, and f1(np,pso,pd) = 0 otherwise. i.e. f1(np,pso,pd)=1 if [np pso pd] is [F F F], [T F T], [F T T] or [T T T], 0 other four conditions samely, &quot;no pressure&quot; = &quot;pump not working&quot; OR &quot;slide-valve closed&quot; OR &quot;pipe leaks&quot;, we get f2 finaly, global function g = f1*f2
11. 11. <ul><li>A Brief Introduction to Graphical Models and Bayesian Networks </li></ul><ul><ul><li>http://www.cs.ubc.ca/~murphyk/Bayes/bnintro.html#appl </li></ul></ul><ul><li>Factor graph applied in decoding, software demos </li></ul><ul><ul><li>http://moser.cm.nctu.edu.tw/html/index.html </li></ul></ul><ul><li>A Noble paper about factor graph & sum-product-algorithm </li></ul><ul><ul><li>http://www.psi.toronto.edu/pubs/2001/frey2001factor.pdf </li></ul></ul>02 More Online Resources ...
12. 12. 02 Thank you for attention TonyOuyang [email_address]
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