A Constraint Propagation Perspective Rina Dechter Bozhena Bidyuk Robert Mateescu Emma Rollon Distributed Belief Propagation Distributed Belief Propagation 1 2 3 4 4 3 2 1 5 5 5 ID: 714242
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Slide1
On the Power of Belief Propagation:A Constraint Propagation Perspective
Rina Dechter
Bozhena
Bidyuk
Robert
Mateescu
Emma
RollonSlide2
Distributed Belief PropagationSlide3
Distributed Belief Propagation
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How many people?Slide4
Distributed Belief Propagation
Causal support
Diagnostic supportSlide5
Belief Propagation in PolytreesSlide6
BP on Loopy GraphsPearl (1988): use of BP to loopy networksMcEliece
, et. Al 1988: IBP’s success on coding networks Lots of research into convergence … and accuracy (?), but:Why IBP works well for coding networksCan we characterize other good problem classesCan we have any guarantees on accuracy (even if converges)Slide7
Arc-consistencySound Incomplete Always converges
(polynomial)A
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21A
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A < D
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B = C
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P(B|A)
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Belief network
Flat constraint network
Flattening the Bayesian NetworkSlide9
A
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Updated belief:
Updated relation:
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Belief Zero Propagation = Arc-ConsistencySlide10
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P(F|B,C)
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Flat Network - ExampleSlide11
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IBP Example – Iteration 1Slide12
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IBP Example – Iteration 2Slide13
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IBP Example – Iteration 3Slide14
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IBP Example – Iteration 4
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1Slide15
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IBP Example – Iteration 5
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1Slide16
IBP – Inference Power for Zero Beliefs
Theorem: Iterative BP performs arc-consistency on the flat network. Soundness:
Inference of zero beliefs by IBP convergesAll the inferred zero beliefs are
correctIncompleteness:
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 empiricallySlide17
Experimental ResultsAlgorithms:IBPIJGP
Measures:Exact/IJGP histogramRecall absolute error
Precision absolute error
Network types:
Coding
Linkage analysis*Grids*Two-layer noisy-OR*CPCS54, CPCS360We investigated empirically if the results for zero beliefs extend to ε-small beliefs (ε > 0)* Instances from the UAI08 competition
Have determinism?YESNOSlide18
Networks with Determinism: Coding
N=200, 1000 instances, w*=15Slide19
Nets w/o Determinism: bn2o
w* = 24w* = 27
w* = 26Slide20
Nets
with Determinism: LinkagePercentage
Absolute Error
pedigree1, w* = 21
Exact Histogram IJGP Histogram Recall Abs. Error Precision Abs. Error
Percentage
Absolute Error
pedigree37, w* = 30
i-bound = 3
i-bound = 7Slide21
Nets with Determinism: Grids
(size, % determinism) =, w* = 22Slide22
Nets w/o Determinism: cpcs
CPCS360: 5 instances, w*=20 CPCS54: 100 instances, w*=15Slide23
The Cutset Phenomena & irrelevant nodesObserved variables break the flow of inference IBP is exact when evidence variables form a cycle-
cutsetUnobserved variables without observed descendents send zero-information to the parent variables – it is irrelevantIn a network without evidence, IBP converges in one iteration top-down
X
X
YSlide24
Nodes with extreme supportObserved variables with xtreme priors or xtreme support can nearly-cut information flow:
D
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Average Error vs. PriorsSlide25
Conclusion: For Networks with DeterminismIBP converges & sound for zero beliefsIBP’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 cutsetIf zeros are cycle-
cutset, all beliefs are exactExtensions to epsilon-cutset were supported empirically
.Slide26
Inference on Trees is Easy and DistributedBelief 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 modelsSlide27
Inference on Poly-Trees is Easy and Distributed