On Distributing a Bayesian Network

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On Distributing a Bayesian Network

• 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
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Step 1 Moralization
GM
G ( V , E )
1. For all w ? V For all u,v?pa(w) add an
edge eu-v. 2. Undirect all edges.
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Step 2 Triangulation
Add edges to GM such that there is no cycle with
length ? 4 that does not contain a chord.
NO
YES
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Bayesian Network G ( V , E )
Moral graph GM
Triangulated graph GT
a
abd
ace
ad
ae
ce
ade
ceg
e
e
eg
de
e
seperators
egh
def
e
Cliques
e.g. ceg ? egh eg
Junction graph GJ (not complete)
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• There are several methods to find MST.
• Kruskals algorithm choose successively a link
  of
• maximal weight unless it creates a cycle.

Junction tree GJT
Junction graph GJ (not complete)
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GJT
In JT cliques becomes vertices
sepsets
Ex ceg ? egh eg
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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
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Global Propagation
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
Cant communicate here!
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Issue MSBN has a tree structure
Cant 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

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
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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
BC
BCE
ABC
BE
E
B
BDE
DEF
DE
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Mysterious convergence
C
CE
ABC
E
E
B
BDE
DEF
DE