1. The junction tree algorithm is used to perform inference in graphical models. It involves constructing a junction tree from a graph, where cliques in the graph become nodes in the tree. 2. Message passing is then performed on the junction tree to calculate marginal and joint probabilities. Messages are passed from parent to child nodes involving updating beliefs based on the intersection of variable sets. 3. Common message passing algorithms include Shafer-Shenoy and Lauritzen-Spiegelhalter, which differ in how messages are computed and passed between nodes.

1. 4.4 Coping with loops

2. • DAG
•
oscillation:
•
•
coherent
• coherent
A
B C
E
D

3. 1. clustering
–
2. conditioning
–
3. stochastic simulation
–

4. 4.4.1 Clustering Methods

5. A
• B C
Z B C
z ∈ {(+b,+c), (¬b,+c), (+b,¬c), (¬b,¬c)} E
D
•
• Z B C A
P(b,c|a)
Z
B,C P(e|c)
E
D
P(d|b,c)

6. •
– (= )
–
•
•
join trees(=clique trees or junction trees)

7. Join Tree
1. NB G
–
–
– → G NB I-map I(G)⊆I(NB)
2. G chordal supergraph G’
※3.2.4 graph-traiangulation algorithm
3. G’ clique join tree T
4. NB
– T clique-node
– clique-node λ
5. T clique-node
NB

8. G Join Tree
1. fill-in G’
1. maximum cardinality search
• 1 |V|( )
•
•
2. for (n=|V| to 1) n
2. G’
3.
C1, C2, ..., Ct
4. Ci Cj(i>j)

9. Join Tree Z1
A P(d|b,c) A,B,C
A P(e|c)
Z2
Z3
B C B C B,C,D C,E
E D λD(z2) λE(z3)
D E D=0 E=1
M zi |z j = P(zi | z j ) = P(zi | zi ∩ z j ). M z2 |z1 = P(z2 | z1 ) = P(b, c, d | b, c) = P(d | b, c)
M z3|z1 = P(z3 | z1 ) = P(c, e | c) = P(e | c)

 0 if z2 = (b, c, +d) π (z1 ) = π z (z1 ) = π z (z1 ) = P(z1 ) = P(a, b, c) = P(b | a)P(c | a)P(a)
λD (z2 ) =  2 3

 1 if z2 = (b, c,¬d) π (z2 ) = ∑ P(z2 | z1 )P(z1 ) = ∑ P(d | b, c)P(a, b, c) = P(d | b, c)P(b, c)

 0 if z3 = (c,¬e)
Z1 a
λE (z3 ) =  π (z3 ) = ∑ P(z3 | z1 )P(z1 ) = ∑ P(e | c)P(a, b, c) = P(e | c)P(c)

 1 if z3 = (c,+e) Z1 a,b
λZ (z1 ) = M z |z • λD (z2 ) = P(¬d | b, c)
2 2 1
BEL(z1 ) = αλZ2 (z1 )λZ3 (z1 )π (z1 )
λZ (z1 ) = M z |z • λE (z3 ) = P(+e | c)
3 3 1
= α P(¬d | b, c)P(+e | c)P(b | a)P(c | a)P(a)
8 BEL(a),BEL(b),BEL(c)

10. 4.4.2 The Method of Conditioning(Reasoning by Assumptions)

11. Conditioning
A •
– a=0 a=1
•
B C
• a
E • BEL(b),BEL(c)
D a
A=0 A=0 A=1 A=1 BEL(b) = P (b | e) = ∑ P(b | a, e) P (a | e)
a
A A A A
B C B C
D=0 D E D=0 D E
E=1 E=1
w0 w1
w = (w1, w0 ) = P(a)
w E = P(a | +e) = α P(+e | a)P(a)
w E,D = P(a | +e,¬d) = α P(¬d | a, +e)P(a | +e)

12. A Short Course on Graphical Models
3 The Junction Tree Algoritms – Mark Paskin
http://ai.stanaford.edu/~paskin/gm-short-course/lec3.pdf

13. Join Tree? ,Junction Tree?
•
en.wikipedia.org/wiki/Tree_decomposition
• Join Tree DAG
• Junction Tree Algorithm

14. Junction Tree
G d T {b,d}
b
f {b}
a
e
{b,e}
{b,c}
c {a,b,c} {b,c,e} {b,e,f}
T G Junction Tree
• singly connected( ):
• covering:
G A C A⊆C
• running intersection:
B C i B C i
※Junction Tree Algorithm

15. Junction Tree
※ BEL
λ π
•
• (
)
•

16. message passing
• message passing protocol: B C
C
• (
1. Collect(C): for (B <- C.children) { Collect(B) }
2. Distribute(C): for (B <- C.children) {Distribute(B)}
• The Shafer-Shenoy Algorithm:
B C
– C
– C
• {b,d}
– Lauritzen-Spiegelhalter
– Hugin 1{b} 4
2 1
{b,e}
{b,c}
{a,b,c} {b,c,e} {b,e,f}
3 4