1. Iterative Join Graph Propagation
Rina Dechter
Kalev Kask
Robert Mateescu
I am going to talk about Iterative Join Graph Propagation, which is work done by RD, KK and myself
UAI 2002
University of California - Irvine

2. What is IJGP?
IJGP is an approximate algorithm for belief updating in Bayesian networks
IJGP is a version of join-tree clustering which is both anytime and iterative
IJGP applies message passing along a join-graph, rather than a join-tree
Empirical evaluation shows that IJGP is almost always superior to other approximate schemes (IBP, MC)
Probabilistic reasoning using Bayesian betworks is known to be a hard problem. Although exact algorithms are available for solving this task, they are most of the time infeasible in practice, due to the large size and connectivity of real life networks.
Approximate schemes are therefore of crucial importance.
We propose here a new algorithm, called IJGP - iterative join graph propagation, which applies message passing along a join-graph.
It is both anytime and iterative, providing the user adjustable tradeoffs between accuracy and time.
The empirical evaluation we conducted was extremely encouraging, showing that IJGP was better than other existing approximate algorithms on several classes of problems.

3. Outline IBP - Iterative Belief Propagation MC - Mini Clustering
IJGP - Iterative Join Graph Propagation
Empirical evaluation
Conclusion
Our new algorithm, IJGP, builds on two existing approximate schemes: IBP, which is an iterative algorightm, on one hand, and a recently introduced scheme, Mini-Clustering, which is an anytime algorithm on the other hand.
We will quickly review the main characteristics of these two algorithms and then present IJGP.

4. Iterative Belief Propagation - IBP
Belief propagation is exact for poly-trees
IBP - applying BP iteratively to cyclic networks
No guarantees for convergence
Works well for many coding networks U1 U2 U3
One step:
update BEL(U1) X1 X2

5. p(d|b), h(1,2)(b,c), p(f|c,d)
Mini-Clustering - MC
Cluster Tree Elimination
Mini-Clustering, i=3
A B C
p(a), p(b|a), p(c|a,b) 1 BC
B C D C D F
p(d|b), h(1,2)(b,c) p(f|c,d)
B C D F
p(d|b), h(1,2)(b,c), p(f|c,d)
B C D F
p(d|b), p(f|c,d) 2 2 BF
sep(2,3) = {B,F}
elim(2,3) = {C,D} A B
B E F
p(e|b,f) 3 C D E EF F
E F G
p(g|e,f) 4 G

6. CTE vs. MC 1 1 2 2 3 3 4 4 Cluster Tree Elimination
ABC
ABC 1 1 BC BC 2
BCDF 2
BCDF BF BF 3
BEF 3
BEF EF EF 4
EFG 4
EFG
Cluster Tree Elimination
Mini-Clustering, i=3

7. IJGP - Motivation IBP is applied to a loopy network iteratively
not an anytime algorithm
when it converges, it converges very fast
MC applies bounded inference along a tree decomposition
MC is an anytime algorithm controlled by i-bound
IJGP combines:
the iterative feature of IBP
the anytime feature of MC

8. IJGP - The basic idea Apply Cluster Tree Elimination to any join-graph
We commit to graphs that are minimal I-maps
Avoid cycles as long as I-mapness is not violated
Result: use minimal arc-labeled join-graphs

9. IJGP - Example A B C A C A ABC C A AB BC C D E ABDE BCE BE C DE CE F
CDEF G H F
FGH H H F FG GH H I J GI
FGI
GHIJ
a) Belief network
a) The graph IBP works on

10. Arc-minimal join-graph
ABC C A
ABC C A AB BC C AB BC
ABDE
BCE
ABDE
BCE BE C C DE CE DE CE
CDEF
CDEF H F
FGH H
FGH H H F F FG GH H FG GH GI GI
FGI
GHIJ
FGI
GHIJ

11. Minimal arc-labeled join-graph
ABC C A
ABC C AB BC AB BC
ABDE
BCE
ABDE
BCE C C DE CE DE CE
CDEF
CDEF
FGH H H
FGH H H F F FG GH F GH GI GI
FGI
GHIJ
FGI
GHIJ

12. Join-graph decompositions
ABC C AB BC BC BC
ABDE
BCE
ABCDE
BCE
ABCDE
BCE C DE CE
CDE CE DE CE
CDEF
CDEF
CDEF
FGH H H
FGH
FGH F F F F GH F GH F GH GI GI GI
FGI
GHIJ
FGI
GHIJ
FGI
GHIJ
a) Minimal arc-labeled join graph
b) Join-graph obtained by collapsing nodes of graph a)
c) Minimal arc-labeled join graph

13. Tree decomposition BC ABCDE BCE ABCDE DE CE CDE CDEF CDEF FGH F F F GHGI
GHI
FGI
GHIJ
FGHI
GHIJ
a) Minimal arc-labeled join graph
a) Tree decomposition

14. Join-graphs more accuracy less complexity A ABDE FGI ABC BCE GHIJ CDEF
FGH C H AB BC BE DE CE F FG GH GI A
ABDE
FGI
ABC
BCE
GHIJ
CDEF
FGH C H AB BC DE CE F GH GI
ABCDE
FGI
BCE
GHIJ
CDEF
FGH BC DE CE F GH GI
ABCDE
FGHI
GHIJ
CDEF
CDE F
GHI
more accuracy
less complexity

15. Minimal arc-labeled decompositionBC BC
ABCDE
BCE
ABCDE
BCE
CDE CE DE CE
CDEF
CDEF
a) Fragment of an
arc-labeled join-graph
a) Shrinking labels to make it a minimal arc-labeled join-graph
Use a DFS algorithm to eliminate cycles relative to each variable

16. p(a), p(c), p(b|ac), p(d|abe),p(e|b,c) h(3,1)(bc)
Message propagation
ABCDE BC
BCE
ABCDE
p(a), p(c), p(b|ac), p(d|abe),p(e|b,c) h(3,1)(bc)
h(3,1)(bc)
CDE 1
BCD 3 CE BC
CDEF
FGH
h(1,2)
CDE CE F F GH 2
CDEF GI
FGI
GHIJ
Minimal arc-labeled:
sep(1,2)={D,E}
elim(1,2)={A,B,C}
Non-minimal arc-labeled:
sep(1,2)={C,D,E}
elim(1,2)={A,B}

17. Bounded decompositions
We want arc-labeled decompositions such that:
the cluster size (internal width) is bounded by i (the accuracy parameter)
the width of the decomposition as a graph (external width) is as small as possible
Possible approaches to build decompositions:
partition-based algorithms - inspired by the mini-bucket decomposition
grouping-based algorithms

18. Partition-based algorithmsF C D B A
G: (GFE)
E: (EBF) (EF)
F: (FCD) (BF)
D: (DB) (CD)
C: (CAB) (CB)
B: (BA) (AB) (B)
A: (A)
GFE
P(G|F,E) EF
EBF
P(E|B,F)
P(F|C,D) F BF
FCD BF CD
CDB
P(D|B) CB B
CAB
P(C|A,B) BA BA
P(B|A) A A
P(A)
a) schematic mini-bucket(i), i=3 b) arc-labeled join-graph decomposition

19. IJGP properties
IJGP(i) applies BP to min arc-labeled join-graph, whose cluster size is bounded by i
On join-trees IJGP finds exact beliefs
IJGP is a Generalized Belief Propagation algorithm (Yedidia, Freeman, Weiss 2001)
Complexity of one iteration:
time: O(deg•(n+N) •d i+1)
space: O(N•d)

20. Empirical evaluation Algorithms: Measures:
Exact
IBP MC
IJGP
Measures:
Absolute error
Relative error
Kullback-Leibler (KL) distance
Bit Error Rate
Time
Networks (all variables are binary):
Random networks
Grid networks (MxM)
CPCS 54, 360, 422
Coding networks

21. Random networks - KL at convergence
This is a comment
evidence=0
evidence=5

22. Random networks - KL vs. iterations
evidence=0
evidence=5

23. Random networks - Time

24. Grid 81 – KL distance at convergence

25. Grid 81 – KL distance vs. iterations

26. Grid 81 – Time vs. i-bound

27. N=400, P=4, 500 instances, 30 iterations, w*=43
Coding networks
N=400, P=4, 500 instances, 30 iterations, w*=43

28. Coding networks - BER
sigma=.22
sigma=.32
sigma=.51
sigma=.65

29. Coding networks - Time

30. CPCS 422 – KL distance
evidence=0
evidence=30

31. CPCS 422 – KL vs. iterations
evidence=0
evidence=30

32. CPCS 422 – Relative error
evidence=30

33. CPCS 422 – Time vs. i-bound
evidence=0

34. Conclusion
IJGP borrows the iterative feature from IBP and the anytime virtues of bounded inference from MC
Empirical evaluation showed the potential of IJGP, which improves with iteration and most of the time with i-bound, and scales up to large networks
IJGP is almost always superior, often by a high margin, to IBP and MC
Based on all our experiments, we think that IJGP provides a practical breakthrough to the task of belief updating