1. Clique Tree & Independence
Inference
Probabilistic
Graphical
Models
Message Passing
Clique Tree & Independence

2. RIP and Independence For an edge (i,j) in T, let:
W<(i,j) = all variables that appear only on Ci side of T
W<(j,i) = all variables that appear only on Cj side
Variables on both sides are in the sepset Si,j
Theorem: T satisfies RIP if and only if, for every (i,j)

3. RIP and Independence 1: C,D 2: G,I,D 3: G,S,I 5: H,G,J 4: G,J,S,L D

4. RIP and Independence
Theorem: T satisfies RIP if and only if, for every edge (i,j) C D I S G L J H 1:
C,D 2:
G,I,D 3:
G,S,I 5:
H,G,J 4:
G,J,S,L D
G,I
G,S
G,J

5. Implications
Each sepset needs to separate graph into two conditionally independent parts
A1,1
A1,2
A1,3
A1,4
A2,1
A2,2
A2,3
A2,4
A3,1
A3,2
A3,3
A3,4
A4,1
A4,2
A4,3
A4,4 A1 A2 Ak … B1 B2 Bm

6. Summary
Correctness of clique tree inference relies on running intersection property
Running intersection property implies separation in original distribution
Implies minimal complexity incurred by any clique tree:
Related to minimal induced width of graph

7. END END END