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Junction Tree Construction. Practice exercise. Example 1: Junction Tree Construction . Step1: Insert link between parents of common child. A. B. D. E. C. G. F. H. Example 1: Junction Tree Construction. Step2: C onvert into undirected graph. A. B. A. B. D. E. D. E. C. C. G.

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Junction tree construction

Junction Tree Construction

Practice exercise


Example 1 junction tree construction
Example 1: Junction Tree Construction

Step1: Insert link between parents of common child

A

B

D

E

C

G

F

H


Example 1 junction tree construction1
Example 1: Junction Tree Construction

Step2: Convert into undirected graph

A

B

A

B

D

E

D

E

C

C

G

G

F

H

F

H


Example 1 junction tree construction2
Example 1: Junction Tree Construction

Step3: Triangulate a graph if it has circle of length>3

A

B

D

E

C

G

F

H

Note : No need of triangulation in this example.


Example 1 junction tree construction3
Example 1: Junction Tree Construction

Step3: Numbered the node if its numbered neighboured nodes are connected together.

1

2

A

B

4

5

3

D

E

C

8

6

7

G

F

H


Example 1 junction tree construction4
Example 1: Junction Tree Construction

Definition:

A Clique in a graph is set of adjacent vertices which is a complete graph.

  • Five cliques are found in Example 1:

    • One clique of four nodes

    • Four cliques of three nodes

Step4: Ordered the node by max cardinality search.

ABCDE

4

1

ACD

5

2

A

B

CDF

6

4

5

3

D

E

C

CEH

7

8

6

7

G

EHG

8

F

H


Example 1 junction tree construction5
Example 1: Junction Tree Construction

Construct Tree

ABCDE

4

ACD

5

ABCDE

CDF

6

CEH

CEH

7

ACD

EHG

8

CDF

EHG


Example 2 junction tree construction
Example 2: Junction Tree Construction

Step1: Insert link between parents of common child

A

B

C

D

E

F

G

H

I


Example 2 junction tree construction1
Example 2: Junction Tree Construction

Step2: Convert into undirected graph

A

A

B

C

D

B

C

D

E

F

G

E

F

G

H

I

H

I


Example 2 junction tree construction2
Example 2: Junction Tree Construction

Step3: Triangulate a graph if it has circle of length>3

A

B

C

D

E

F

G

H

I


Example 2 junction tree construction3
Example 2: Junction Tree Construction

Step3: Numbered the node if its numbered neighboured nodes are connected together.

1

A

2

4

3

F has three numbered neighbour B, C and D where B and D are not connected therefore add link between B and D and restart numbering again from beginning.

B

C

D

5

E

F

G

H

I


Example 2 junction tree construction4
Example 2: Junction Tree Construction

Step3: Numbered the node if its numbered neighboured nodes are connected together.

1

A

2

4

3

B

C

D

5

6

7

E

F

G

H

I

8

9


Example 2 junction tree construction5
Example 2: Junction Tree Construction

  • Five cliques are found in Example 2:

    • Two cliques of four nodes

    • Four cliques of three nodes

Step4: Ordered the node by max cardinality search.

Clique Nodes

Max Cardinality

1

A

ABCD

4

BCDF

5

2

4

3

B

C

D

BEF

6

DFG

5

7

6

7

E

F

G

EFH

8

H

I

FGI

8

9

9


Example 2 junction tree construction6
Example 2: Junction Tree Construction

Construct Tree

ABCD

4

ABCD

BCDF

5

BEF

6

BCDF

DFG

7

DFG

EFH

8

BEF

FGI

9

FGI

EFH


Example 2 junction tree construction with different triangulation
Example 2: Junction Tree Construction (with different triangulation )

Step1: Insert link between parents of common child

A

B

C

D

E

F

G

H

I


Example 2 junction tree construction7
Example 2: Junction Tree Construction triangulation )

Step2: Convert into undirected graph

A

A

B

C

D

B

C

D

E

F

G

E

F

G

H

I

H

I


Example 2 junction tree construction8
Example 2: Junction Tree Construction triangulation )

Step3: Triangulate a graph if it has circle of length>3

A

B

C

D

E

F

G

H

I


Example 2 junction tree construction9
Example 2: Junction Tree Construction triangulation )

Step3: Numbered the node if its numbered neighboured nodes are connected together.

1

A

2

4

3

G has three numbered neighbour F, C and D where F and D are not connected therefore add link between F and D and restart numbering again from beginning.

B

C

D

6

5

7

E

F

G

H

I


Example 2 junction tree construction10
Example 2: Junction Tree Construction triangulation )

Step3: Numbered the node if its numbered neighboured nodes are connected together.

1

A

2

4

3

F has three numbered neighbour E, C and D where E and D are not connected therefore add link between E and D and restart numbering again from beginning.

B

C

D

6

5

E

F

G

H

I


Example 2 junction tree construction11
Example 2: Junction Tree Construction triangulation )

Step3: Numbered the node if its numbered neighboured nodes are connected together.

1

A

2

4

3

E has three numbered neighbour B, C and D where B and D are not connected therefore add link between B and D and restart numbering again from beginning.

B

C

D

5

E

F

G

H

I


Example 2 junction tree construction12
Example 2: Junction Tree Construction triangulation )

Step3: Numbered the node if its numbered neighboured nodes are connected together.

1

A

2

4

3

B

C

D

6

5

7

E

F

G

H

8

I

9


Example 2 junction tree construction13
Example 2: Junction Tree Construction triangulation )

  • Cliques are found in Example 2:

    • Four cliques of four nodes.

    • Two cliques of three nodes.

Step4: Ordered the node by max cardinality search.

1

A

Clique Nodes

Max Cardinality

2

4

3

ABCD

4

B

C

D

BCDE

5

6

CDEF

5

6

7

E

F

G

CDFG

7

EFH

8

H

8

I

9

FGI

9


Example 2 junction tree construction14
Example 2: Junction Tree Construction triangulation )

Construct Tree

ABCD

4

ABCD

BCDE

5

CDEF

6

BCDE

CDFG

7

CDEF

EFH

8

FGI

9

EFH

CDFG

FGI


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