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On Distributing a Bayesian On Distributing a Bayesian NetworkNetwork

Thor Whalen

Metron, Inc.

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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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Outline

• Need a Review of JT?

• Distributing a BN using the JT: Issues.

• Directions for Solutions

• Matlab: New functionality and experiments

• And now what?

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Secondary Structure/Junction Tree• multi-dim. random variables• joint probabilities (potentials)

Bayesian Network• one-dim. random variables• conditional probabilities

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Building a Junction Tree

DAG

Moral Graph

Triangulated Graph

Junction Tree

Identifying Cliques

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Step 1: Moralization

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1. For all w V:• For all u,vpa(w) add an edge e=u-v.

2. Undirect all edges.

GMG = ( V , E )

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Step 2: Triangulation

Add edges to GM such that there is no cyclewith length 4 that does not contain a chord.

NO YES

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GM GT

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Bayesian NetworkG = ( V , E )

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Moral graph GM Triangulated graph GT

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Junction graph GJ (not complete)e.g. ceg egh = eg Cliques

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Junction graph GJ (not complete)

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Junction tree GJT

There are several methods to find MST.

Kruskal’s algorithm: choose successively a link of

maximal weight unless it creates a cycle.

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In JT cliquesbecomesvertices

GJT

Ex: ceg egh = eg

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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

Propagating potentialsPropagating potentials

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1. COLLECT-EVIDENCE messages 1-52. DISTRIBUTE-EVIDENCE messages 6-10

Global PropagationGlobal Propagation

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Root

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Using the JT for the MSBN

• How?

• Issue: Clique Granularity

• Issue: Clique Association

• Issue: MSBN must have tree structure

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Take a JT

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Partition JT into sub-trees

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Issue: Clique Granularity

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Issue: Clique Granularity

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Issue: Clique association

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Issue: Clique association

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Issue: MSBN has a tree structure

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Issue: MSBN has a tree structure

Can’t communicate here!

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Issue: MSBN has a tree structure

Can’t communicate here!

Though these two subnets may share many variables.

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Directions for Solutions

• Controlling Granularity

• Increasing Granularity

• Choosing the JT

• Alternative communication graphs

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Controlling Granularity

• Moralization and Triangulation → Cliques

• Moralization: No choice.

• Triangulation: Some choice.

• Triangulate keeping the subnets in mind.

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Increasing Granularity

• Can we break down cliques?

Conjecture:Given two sets of random variables X and Y, let

dist(P(X-Y|Y),P(X-Y|X Y))(X,Y) =

vol(X Y)

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.

Z X Y

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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.

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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…

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MatLab: The new Stuff:

• Netica can talk to the Matlab tool.

• View/Interact tool is nicer

• JG for communication graph… (or not)

• Query cliques

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Propagating with cycles

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Mysterious convergence

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