On the Power of Belief Propagation: A Constraint Propagation Perspective Rina Dechter Bozhena Bidyuk Robert Mateescu Emma Rollon

Distributed Belief Propagation

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Distributed Belief Propagation Causal support Diagnostic support

Belief Propagation in Polytrees

BP on Loopy Graphs Pearl (1988): use of BP to loopy networks McEliece, et. Al 1988: IBP’s success on coding networks Lots of research into convergence … and accuracy (?), but: Why IBP works well for coding networks Can we characterize other good problem classes Can we have any guarantees on accuracy (even if converges)

Arc-consistency Sound Incomplete Always converges (polynomial) AB CD A B D C < < < = A < B A < D D < C B = C

A ABABACAC ABDBCF DFG B B D F A A A C 1 AP(A) …0 ABP(B|A) ……0 ABDP(D|A,B) ………0 DFGP(G|D,F) ………0 BCFP(F|B,C) ………0 ACP(C|A) ……0 A AB ABD DFG BCF AC A ABABACAC ABDBCF DFG B B D F A A A C 1 Belief network Flat constraint network Flattening the Bayesian Network

ABP(B|A) 12> ……0 AB Ah 1 2 (A) 1>0 2 3 …0 Bh 1 2 (B) 1>0 2 3 …0 Bh 1 2 (B) 1>0 3 …0 A B B 1 3 Updated belief: Updated relation: AB Bel (A,B) 13> ……0 AB A ABAC ABDBCF DFG B B D F A A A C 1 Belief Zero Propagation = Arc-Consistency

AP(A) …0 ACP(C|A) ……0 ABP(B|A) ……0 BCFP(F|B,C) ………0 ABDP(D|A,B) ………0 DFGP(G|D,F) ………0 A ABABACAC ABDBCF DFG B B D F A A A C 1 Flat Network - Example

AP(A) 1>0 3 …0 ACP(C|A) ……0 ABP(B|A) > ……0 BCFP(F|B,C) ………0 ABDP(D|A,B) ………0 DFGP(G|D,F) 2131 ………0 A ABABACAC ABDBCF DFG B B D F A A A C 1 IBP Example – Iteration 1

AP(A) 1>0 3 …0 ACP(C|A) ……0 ABP(B|A) ……0 BCFP(F|B,C) 3211 ………0 ABDP(D|A,B) ………0 DFGP(G|D,F) 2131 ………0 A ABABACAC ABDBCF DFG B B D F A A A C 1 IBP Example – Iteration 2

AP(A) 1>0 3 …0 ACP(C|A) ……0 ABP(B|A) 131 ……0 BCFP(F|B,C) 3211 ………0 ABDP(D|A,B) ………0 DFGP(G|D,F) 2131 ………0 A ABABACAC ABDBCF DFG B B D F A A A C 1 IBP Example – Iteration 3

AP(A) 11 …0 ACP(C|A) ……0 ABP(B|A) 131 ……0 BCFP(F|B,C) 3211 ………0 ABDP(D|A,B) 1321 ………0 DFGP(G|D,F) 2131 ………0 IBP Example – Iteration 4 A ABABACAC ABDBCF DFG B B D F A A A C 1

AP(A) 11 …0 ACP(C|A) 121 ……0 ABP(B|A) 131 ……0 BCFP(F|B,C) 3211 ………0 ABDP(D|A,B) 1321 ………0 DFGP(G|D,F) 2131 ………0 ABCDFGBelief ………………0 IBP Example – Iteration 5 A ABABACAC ABDBCF DFG B B D F A A A C 1

IBP – Inference Power for Zero Beliefs Theorem: Iterative BP performs arc-consistency on the flat network. Soundness: Inference of zero beliefs by IBP converges All the inferred zero beliefs are correct Incompleteness: Iterative BP is as weak and as strong as arc-consistency Continuity Hypothesis: IBP is sound for zero -  IBP is accurate for extreme beliefs? Tested empirically

Experimental Results Algorithms: IBP IJGP Measures: Exact/IJGP histogram Recall absolute error Precision absolute error Network types: Coding Linkage analysis* Grids* Two-layer noisy-OR* CPCS54, CPCS360 We investigated empirically if the results for zero beliefs extend to ε-small beliefs (ε > 0) * Instances from the UAI08 competition Have determinism? YES NO

Networks with Determinism: Coding N=200, 1000 instances, w*=15

Nets w/o Determinism: bn2o w* = 24 w* = 27 w* = 26

Nets with Determinism: Linkage Percentage Absolute Error pedigree1, w* = 21 Exact Histogram IJGP Histogram Recall Abs. Error Precision Abs. Error Percentage Absolute Error pedigree37, w* = 30 i-bound = 3i-bound = 7

Nets with Determinism: Grids (size, % determinism) =, w* = 22

Nets w/o Determinism: cpcs CPCS360: 5 instances, w*=20 CPCS54: 100 instances, w*=15

The Cutset Phenomena & irrelevant nodes Observed variables break the flow of inference IBP is exact when evidence variables form a cycle-cutset Unobserved variables without observed descendents send zero- information to the parent variables – it is irrelevant In a network without evidence, IBP converges in one iteration top-down X X Y

Nodes with extreme support Observed variables with xtreme priors or xtreme support can nearly-cut information flow: D B1B1 EDED A B2B2 EBEB ECEC EBEB ECEC C1C1 C2C2 Average Error vs. Priors

Conclusion: For Networks with Determinism IBP converges & sound for zero beliefs IBP’s power to infer zeros is as weak or as strong as arc-consistency However: inference of extreme beliefs can be wrong. Cutset property (Bidyuk and Dechter, 2000): Evidence and inferred singleton act like cutset If zeros are cycle-cutset, all beliefs are exact Extensions to epsilon-cutset were supported empirically.

Inference on Trees is Easy and Distributed Belief updating (sum-prod) MPE (max-prod) CSP – consistency (projection-join) #CSP (sum-prod) P(X) P(Y|X)P(Z|X) P(T|Y) P(R|Y)P(L|Z)P(M|Z) Trees are processed in linear time and memory Also Acyclic graphical models

Inference on Poly-Trees is Easy and Distributed