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On the Power of Belief Propagation: A Constraint Propagation PerspectivePowerPoint Presentation

On the Power of Belief Propagation: A Constraint Propagation Perspective

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### On the Power of Belief Propagation:A Constraint Propagation Perspective

Rina Dechter

BozhenaBidyuk

RobertMateescu

EmmaRollon

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)

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B

1

1

2

2

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3

3

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D

C

D

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1

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4

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ABD

BCF

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DFG

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DFG

Flattening the Bayesian NetworkFlat constraint network

Belief network

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5

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Belief Zero Propagation = Arc-ConsistencyUpdated relation:

Updated belief:

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

IBP

IJGP

Experimental ResultsWe investigated empirically if the results for zero beliefs extend to ε-small beliefs (ε > 0)

- Measures:
- Exact/IJGP histogram
- Recall absolute error
- Precision absolute error

- Network types:
- Coding
- Linkage analysis*
- Grids*
- Two-layer noisy-OR*
- CPCS54, CPCS360

YES

Have determinism?

NO

* Instances from the UAI08 competition

Networks withDeterminism: Coding

N=200, 1000 instances, w*=15

NetswithDeterminism: Linkage

Exact Histogram IJGP Histogram Recall Abs. Error Precision Abs. Error

pedigree1, w* = 21

Absolute Error

Percentage

pedigree37, w* = 30

Absolute Error

Percentage

i-bound = 3

i-bound = 7

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

Y

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Nodes with extreme support

Observed variables with xtreme priors or xtreme support can nearly-cut information flow:

A

B1

C1

EB

EC

B2

C2

D

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ED

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.

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