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Transcript of Intro to Junction Tree propagation and adaptations for a Distributed Environment Thor Whalen Metron,...
![Page 1: Intro to Junction Tree propagation and adaptations for a Distributed Environment Thor Whalen Metron, Inc.](https://reader036.fdocuments.net/reader036/viewer/2022062309/5697bf7b1a28abf838c83ad9/html5/thumbnails/1.jpg)
Intro to Junction Tree Intro to Junction Tree propagation and propagation and
adaptations for a Distributed adaptations for a Distributed EnvironmentEnvironment
Thor Whalen
Metron, Inc.
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8conflict
This naive approach of updating the network inherits oscillation problems!
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Idea behind the Junction Tree Idea behind the Junction Tree AlgorithmAlgorithm
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clever
algorithm
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Secondary Structure/Junction Tree• multi-dim. random variables• joint probabilities (potentials)
Bayesian Network• one-dim. random variables• conditional probabilities
abd
ade
ace
ceg
eghdef
ad ae ce
de eg
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• Write query in the form
• Iteratively– Move all irrelevant terms outside of innermost sum– Perform innermost sum, getting a new term– Insert the new term into the product
3 2
1( ) ( | ( ))k
i iX X X i
P X P X par X
Variable EliminationVariable Elimination (General Idea) (General Idea)
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Eaxmple of Variable Elimination
• The “Asia” network:
Visit to Asia
Smoking
Lung CancerTuberculosis
Abnormalityin Chest
Bronchitis
X-Ray Dyspnea
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V S
LT
A B
X D
),|( )|( ),|( )|( )|( )|( )( )( badPaxPltaPsbPslPvtPsPvP
We are interested in P(d)
- Need to eliminate: v,s,x,t,l,a,b
Initial factors:
Brute force:
v s x t l a b
badPaxPltaPsbPslPvtPsPvPdP ),|( )|( ),|( )|( )|( )|( )( )()(
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V S
LT
A B
X D
),|( )|( ),|( )|( )|( )|( )( )( badPaxPltaPsbPslPvtPsPvP
Eliminate variables in order:
Initial factors:
v
v vtPvPtf )|()()(
baltxsv
),|()|(),|()|()|()()( badPaxPltaPsbPslPsPtfv
[ Note: fv(t) = P(t) In general, result of elimination is not necessarily a probability term ]
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),|( )|( ),|( )|( )|( )|( )( )( badPaxPltaPsbPslPvtPsPvP
Eliminate variables in order:
Initial factors:
V S
LT
A B
X D
baltxsv
),|( )|( ),|( )|( )|( )( )( badPaxPltaPsbPslPsPtfv
s
s slPsbPsPlbf )|()|()(),(
),|()|(),|(),()( badPaxPltaPlbftf sv
[ Note: result of elimination may be a function of several variables ]
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),|(),|()(),()( badPltaPaflbftf xsv
),|( )|( ),|( )|( )|( )|( )( )( badPaxPltaPsbPslPvtPsPvP
Eliminate variables in order:
Initial factors:
V S
LT
A B
X D
baltxsv
),|( )|( ),|( )|( )|( )( )( badPaxPltaPsbPslPsPtfv),|( )|( ),|( ),( )( badPaxPltaPlbftf sv
x
x axPaf )|()([ Note: fx(a) = 1 for all values of a ]
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),|( ),|( )( ),( )( badPltaPaflbftf xsv
),|( )|( ),|( )|( )|( )|( )( )( badPaxPltaPsbPslPvtPsPvP
Eliminate variables in order:
Initial factors:
V S
LT
A B
X D
baltxsv
),|( )|( ),|( )|( )|( )( )( badPaxPltaPsbPslPsPtfv),|( )|( ),|( ),( )( badPaxPltaPlbftf sv
t
vt ltaPtflaf ),|()(),(
),|(),()(),( badPlafaflbf txs
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),|( ),|( )( ),( )( badPltaPaflbftf xsv
),|( )|( ),|( )|( )|( )|( )( )( badPaxPltaPsbPslPvtPsPvP
Eliminate variables in order:
Initial factors:
V S
LT
A B
X D
baltxsv
),|( )|( ),|( )|( )|( )( )( badPaxPltaPsbPslPsPtfv),|( )|( ),|( ),( )( badPaxPltaPlbftf sv
),|( ),( )( ),( badPlafaflbf txs
l
tsl laflbfbaf ),(),(),( ),|()(),( badPafbaf xl
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),|( )( ),( badPafbaf xl
),|( ),|( )( ),( )( badPltaPaflbftf xsv
),|( )|( ),|( )|( )|( )|( )( )( badPaxPltaPsbPslPvtPsPvP
Eliminate variables in order:
Initial factors:
V S
LT
A B
X D
baltxsv
),|( )|( ),|( )|( )|( )( )( badPaxPltaPsbPslPsPtfv),|( )|( ),|( ),( )( badPaxPltaPlbftf sv
),|( ),( )( ),( badPlafaflbf txs
a
xla badpafbafdbf ),|()(),(),( ),( dbfa
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),|( )( ),( badPafbaf xl
),|( ),|( )( ),( )( badPltaPaflbftf xsv
),|( )|( ),|( )|( )|( )|( )( )( badPaxPltaPsbPslPvtPsPvP
Eliminate variables in order:
Initial factors:
V S
LT
A B
X D
baltxsv
),|( )|( ),|( )|( )|( )( )( badPaxPltaPsbPslPsPtfv),|( )|( ),|( ),( )( badPaxPltaPlbftf sv
),|( ),( )( ),( badPlafaflbf txs
),( dbfa )(),()( dfdbfdf bb
ab
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Intermediate factors
baltxsv
ga (l, t,d,b, x)
gb (l, t,d, x,s)
gx (l, t,d,s)
gt (l, t,s,v)
gv (l,d,s)
gs(l,d)
gl (d))(),(),(),(
)(),(
)(
dfdbfbaflaf
aflbf
tf
b
a
l
t
x
s
v
lsvtxba In our previous example: With a different ordering:
V S
LT
A B
X D
Complexity is exponential in the size of these factors!
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Notes about variable elimination
• Actual computation is done in the elimination steps
• Computation depends on the order of elimination
• For each query we need to compute everything again!– Many redundant calculations
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Junction Trees
• The junction tree algorithm “generalizes” Variable Elimination to avoid redundant calculations
• The JT algorithm compiles a class of elimination orders into a data structure that supports the computation of all possible queries.
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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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Step 2: Triangulation (cont.)
• Each elimination ordering triangulates the graph, not necessarily in the same way:
A
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Step 2: Triangulation (cont.)
• Intuitively, triangulations with as few fill-ins as possible are preferred– Leaves us with small cliques (small probability tables)
• A common heuristic: Repeat until no nodes remain:
– Find the node whose elimination would require the least number of fill-ins (may be zero).
– Eliminate that node, and note the need for a fill-in edge between any two non-adjacent neighbors.
• Add the fill-in edges to the original graph.
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vertex induced addedremoved clique edges
1 h egh -2 g ceg -3 f def -4 c ace a-e
vertex induced added removed clique edges
5 b abd a-d6 d ade -7 e ae -8 a a -
GT
GM
Eliminate the vertex that requires least number of edges to be added.
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Step 3: Junction Graph
• A junction graph for an undirected graph G is an undirected, labeled graph.
• The nodes are the cliques in G.
• If two cliques intersect, they are joined in the junction graph by an edge labeled with their intersection.
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Bayesian NetworkG = ( V , E )
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Moral graph GM Triangulated graph GT
abd
ade
ace
ceg
eghdef
ad ae ce
de eg
seperators
Junction graph GJ (not complete)e.g. ceg egh = eg Cliques
e
e
e
a
e
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Step 4: Junction Tree
• A junction tree is a sub-graph of the junction graph that – Is a tree – Contains all the cliques (spanning tree)– Satisfies the running intersection property:
for each pair of nodes U, V, all nodes on the path between U and V contain VU
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Running intersection?Running intersection?All vertices C and sepsets S along the path between any
two vertices A and B contain the intersection AB.
abd
ade
ace
ceg
eghdef
ad ae ce
de eg
Ex: A={a,b,d}, B={a,c,e} AB={a}C={a,d,e}{a}, S1={a,d}{a}, S2={a,e}{a}
AB
C
S1 S2
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A few useful Theorems
• Theorem: An undirected graph is triangulated if and only if its junction graph has a junction tree
• Theorem: A sub-tree of the junction graph of a triangulated graph is a junction tree if and only if it is a spanning of maximal weight (sum of number the of variables in the domain of the link).
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Junction graph GJ (not complete)
abd
ade
ace
ceg
eghdef
ad ae ce
de eg
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abd
ade
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ceg
eghdef
ad ae ce
de eg
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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Colorful example
• Compute the elimination cliques(the order here is f, d, e, c, b, a).
• Form the complete junction graph over the maximal elimination cliques and find a maximum-weight spanning tree.
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Principle of Inference
DAG
Junction Tree
Inconsistent Junction Tree
Initialization
Consistent Junction Tree
Propagation
)|( eE vVP
Marginalization
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abd
ade
ace
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ad ae ce
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sepsets
In JT cliquesbecomesvertices
GJT
Ex: ceg egh = eg
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PotentialsPotentials
DEFINITION: A potential A over a set of variables XA is a function that maps each instantiation of xA into a non-negative real number.
Ex: A potential abc over
the set of vertices {a,b,c}.
Xa has four states, and
Xb and Xc has three
states.
A joint probability is a special case
of a potential where A(xA)=1.
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The potentials in the junction tree are not consistent with each other., i.e. if we use marginalization to get the probability distribution for a variable Xu we will get different results depending on which clique we use.
abd
ade
ace
ceg
eghdef
ad ae ce
de eg
P(Xa) = ade
= (0.02, 0.43, 0.31, 0.12)
de
P(Xa) = ace
= (0.12, 0.33, 0.11, 0.03)
ce
The potentials might not even sum to one, so they are not joint probability distributions.
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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
32
5
1 48 10
9
7
6
Root
abd
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ad ae ce
de eg
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A priori distributionA priori distribution
global propagation
potentials are consistent
Marginalizations gives probability distributions for the variables
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Example: Create Join Tree
B C
A D
(this BN corresponds to an HMM with 2 time steps:
Junction Tree:
B,CA,B C,DB C
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Example: Initialization
VariableAssociated
Cluster Potential function
A A,B
B A,B
C B,C
D C,D
, ( )A B P B
, ( ) ( | )A B P B P A B
, ( | )B C P C B
, ( | )C D P D C
B,CA,B C,DB C
B C
A D
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Example: Collect Evidence
• Choose arbitrary clique, e.g. B,C, where all potential functions will be collected.
• Call recursively neighboring cliques for messages:
• 1. Call A,B:– 1. Projection onto B:
– 2. Absorption:, ( ) ( | ) ( )B A B
A A
P B P A B P B
, , ( | ) ( ) ( , )BB C B C old
B
P C B P B P B C
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Example: Collect Evidence (cont.)
• 2. Call C,D:– 1. Projection:
– 2. Absorption:
, ( | ) 1C C DD D
P D C
, , ( , )CB C B C old
C
P B C
B,CA,B C,DB C
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Example: Distribute Evidence
• Pass messages recursively to neighboring nodes
• Pass message from B,C to A,B:– 1. Projection:
– 2. Absorption:, ( , ) ( )B B C
C C
P B C P B
, ,
( )( , )
( )B
A B A B oldB
P BP A B
P B
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Example: Distribute Evidence (cont.)
• Pass message from X1,X2 to X2,Y2:– 1. Projection:
– 2. Absorption:
, ( , ) ( )C B CB B
P B C P C
, ,
( )( | ) ( , )
1C
C D C D oldC
P CP D C P C D
B,CA,B C,DB C
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Netica’s Animal Characteristics BN
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Subnet 1:
* JTnode 1: An,En
* JTnode 2: An,Sh
* JTnode 3: An,Cl
Subnet 2:
* JTnode 4: Cl,Yo
* JTnode 5: Cl,Wa
Subnet 3:
* JTnode 6: Cl,Bod