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On Distributing a Bayesian Network

On Distributing a Bayesian Network. Thor Whalen Metron, Inc. Major discussion points. Functionality afforded by old code Issues and problems with old approach Functionality trying to add Functionality successfully added What is not yet working correctly What sill have to try to add.

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On Distributing a Bayesian Network

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  1. On Distributing a Bayesian Network Thor Whalen Metron, Inc.

  2. Major discussion points • Functionality afforded by old code • Issues and problems with old approach • Functionality trying to add • Functionality successfully added • What is not yet working correctly • What sill have to try to add

  3. Outline • Need a Review of JT? • Distributing a BN using the JT: Issues. • Directions for Solutions • Matlab: New functionality and experiments • And now what?

  4. ae ce ad b d g h a e c f de eg egh abd ceg ade ace def Secondary Structure/ Junction Tree • multi-dim. random variables • joint probabilities (potentials) Bayesian Network • one-dim. random variables • conditional probabilities

  5. Moral Graph Triangulated Graph Junction Tree Identifying Cliques Building a Junction Tree DAG

  6. b b b g d d g h h g d h a e a a e e c c c f f f Step 1: Moralization G = ( V , E ) GM 1. For all w V: • For all u,vpa(w) add an edge e=u-v. 2. Undirect all edges.

  7. b b g h g h d d e a a e c c f f GM GT Step 2: Triangulation Add edges to GM such that there is no cycle with length  4 that does not contain a chord. NO YES

  8. b b b b g h d d d d g h g g h d g h d a e e a e a e e e e a a e a c c c c c f f f f Bayesian Network G = ( V , E ) Moral graph GM Triangulated graph GT a abd ace ad ae ce ade e ceg e eg de e seperators egh def e Cliques e.g. ceg  egh = eg Junction graph GJ (not complete)

  9. ae ce ad a abd ace ad ae ce de eg egh abd ade ceg ace def ade e ceg e eg de e egh def e There are several methods to find MST. Kruskal’s algorithm: choose successively a link of maximal weight unless it creates a cycle. Junction tree GJT Junction graph GJ (not complete)

  10. ae ce ad b d d g h g d e a e e e e a a c c f de eg abd egh ceg ade ace def GJT In JT cliques becomes vertices sepsets Ex: ceg  egh = eg

  11. Propagating potentials Message Passing from clique A to clique B 1. Project the potential of A into SAB 2. Absorb the potential of SAB into B Projection Absorption

  12. 1. COLLECT-EVIDENCE messages 1-5 2. DISTRIBUTE-EVIDENCE messages 6-10 abd ace 3 2 7 ae ce ad 6 5 9 ade ceg de eg 1 8 4 10 def egh Global Propagation Root

  13. Using the JT for the MSBN • How? • Issue: Clique Granularity • Issue: Clique Association • Issue: MSBN must have tree structure

  14. Take a JT

  15. Partition JT into sub-trees

  16. Issue: Clique Granularity

  17. Issue: Clique Granularity

  18. Issue: Clique association

  19. Issue: Clique association

  20. Issue: MSBN has a tree structure

  21. Issue: MSBN has a tree structure Can’t communicate here!

  22. Issue: MSBN has a tree structure Can’t communicate here! Though these two subnets may share many variables.

  23. Directions for Solutions • Controlling Granularity • Increasing Granularity • Choosing the JT • Alternative communication graphs

  24. Controlling Granularity • Moralization and Triangulation → Cliques • Moralization: No choice. • Triangulation: Some choice. • Triangulate keeping the subnets in mind.

  25. Increasing Granularity • Can we break down cliques? Conjecture: Given two sets of random variables X and Y, let Given a clique Z whose set of random variables Z is covered by sets X and Y. If ∆(X,Y) is small enough then we may replace Z by X and Y. Y X Z

  26. Choosing The JT • Once the cliques have been formed (hence the JG), any maximal weight spanning tree of the JG will do for the JT. • Again, we should choose this tree so as to reduce the overhead when connecting subnets internally.

  27. Alternative communication Graphs We raise the question as to wether it is possible to perform intra- and extra-subnet calibration using some non-tree sub-graph of the JG…

  28. MatLab: The new Stuff: • Netica can talk to the Matlab tool. • View/Interact tool is nicer • JG for communication graph… (or not) • Query cliques

  29. Propagating with cycles BC BCE ABC BE B E BDE DEF DE

  30. Mysterious convergence C CE ABC E B E BDE DEF DE

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