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

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

6. Step 1: Moralization G = ( V , E ) GM 1. For all w  V:b c d e f g h a b c d e f g h a b c d e f g h
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. Step 2: Triangulation GM GTb c d e f g h GM GT
Add edges to GM such that there is no cycle
with length  4 that does not contain a chord. NO
YES

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

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

10. GJT a b d c e f g h abd ade ace ceg egh def ad ae ce de eg
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. Global Propagation 1. COLLECT-EVIDENCE messages 1-5
2. DISTRIBUTE-EVIDENCE messages 6-10
Root
abd
ade
ace
ceg
egh
def ad ae ce de eg 3 2 7 9 6 5 1 8 4 10

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 Z X

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 cyclesBC
BCE
ABC BE B E
BDE
DEF DE

30. Mysterious convergence
ABC E B E
BDE
DEF DE