On the Power of Belief Propagation:A Constraint Propagation Perspective

Rina Dechter

BozhenaBidyuk

RobertMateescu

EmmaRollon

Distributed Belief Propagation

Distributed Belief Propagation

1 2 3 4

4 3 2 1 555 5 5

How many people?

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)

)( 11uX

1U 2U 3U

2X1X

)( 12xU

)( 12uX

)( 13xU

Arc-consistency

Sound Incomplete Always converges

(polynomial)

A B

CD

321A

321B

321D

321C

<

<< =

A < B

1 22 3

A < D1 22 3

D < C1 22 3

B = C

1 12 23 3

A

AB AC

ABD BCF

DFG

B

4 5

3

6

2

B

D F

A

A

A

C

1

A P(A)1 .22 .53 .3… 0

A B P(B|A)1 2 .31 3 .72 1 .42 3 .63 1 .13 2 .9… … 0

A B D P(D|A,B)1 2 3 11 3 2 12 1 3 12 3 1 13 1 2 13 2 1 1… … … 0 D F G P(G|D,F)

1 2 3 12 1 3 1… … … 0

B C F P(F|B,C)1 2 3 13 2 1 1… … … 0

A C P(C|A)1 2 13 2 1… … 0

A123

A B1 21 32 12 33 13 2

A B D1 2 31 3 22 1 32 3 13 1 23 2 1

D F G1 2 32 1 3

B C F1 2 33 2 1

A C1 23 2A

AB AC

ABD BCF

DFG

B

4 5

3

6

2

B

D F

A

A

A

C

1

Belief network Flat constraint network

Flattening the Bayesian Network

j)elim(i, inek

iki

ji

j

hph ))(()}({

)) ( ( )(ikinekil

ji hRh

ij

A B P(B|A)

1 2 >01 3 >02 1 >02 3 >03 1 >03 2 >0… … 0

A B1 21 32 12 33 13 2

A h12(

A)1 >02 >03 >0… 0

21h

24h

25h

B h12(

B)1 >02 >03 >0… 0

B h12(

B)1 >03 >0… 0

A123

21h

24h

25h

B123

B13

Updated belief: Updated relation:

25

24

21)|(),( hhhABPBABel

25

24

21 ),(),( hhhBARBAR

A B Bel (A,B)

1 3 >02 1 >02 3 >03 1 >0… … 0

A B1 32 12 33 1

A

AB AC

ABD BCF

DFG

B

4 5

3

6

2

B

D F

A

A

A

C

1

Belief Zero Propagation = Arc-Consistency

A P(A)

1 .2

2 .5

3 .3

… 0

A C P(C|A)

1 2 1

3 2 1

… … 0

A B P(B|A)

1 2 .3

1 3 .7

2 1 .4

2 3 .6

3 1 .1

3 2 .9

… … 0

B C F P(F|B,C)

1 2 3 1

3 2 1 1

… … … 0

A B D P(D|A,B)

1 2 3 1

1 3 2 1

2 1 3 1

2 3 1 1

3 1 2 1

3 2 1 1

… … … 0D F G P(G|D,F)

1 2 3 1

2 1 3 1

… … … 0

1R

2R

4R

3R

5R

6R

A

AB AC

ABD BCF

DFG

B

4 5

3

6

2

B

D F

A

A

A

C

1

Flat Network - Example

A P(A)

1 >0

3 >0

… 0

A C P(C|A)

1 2 1

3 2 1

… … 0

A B P(B|A)

1 3 1

2 1 >0

2 3 >0

3 1 1

… … 0

B C F P(F|B,C)

1 2 3 1

3 2 1 1

… … … 0

A B D P(D|A,B)

1 3 2 1

2 3 1 1

3 1 2 1

3 2 1 1

… … … 0

D F G P(G|D,F)

2 1 3 1

… … … 0

2R

4R

3R

5R

6R

A

AB AC

ABD BCF

DFG

B

4 5

3

6

2

B

D F

A

A

A

C

1

IBP Example – Iteration 11R

A P(A)

1 >0

3 >0

… 0

A C P(C|A)

1 2 1

3 2 1

… … 0

A B P(B|A)

1 3 1

3 1 1

… … 0

B C F P(F|B,C)

3 2 1 1

… … … 0

A B D P(D|A,B)

1 3 2 1

3 1 2 1

… … … 0

D F G P(G|D,F)

2 1 3 1

… … … 0

2R

4R

3R

5R

6R

A

AB AC

ABD BCF

DFG

B

4 5

3

6

2

B

D F

A

A

A

C

1

IBP Example – Iteration 21R

A P(A)

1 >0

3 >0

… 0

A C P(C|A)

1 2 1

3 2 1

… … 0

A B P(B|A)

1 3 1… … 0

B C F P(F|B,C)

3 2 1 1

… … … 0

A B D P(D|A,B)

1 3 2 1

3 1 2 1

… … … 0

D F G P(G|D,F)

2 1 3 1

… … … 0

2R

4R

3R

5R

6R

A

AB AC

ABD BCF

DFG

B

4 5

3

6

2

B

D F

A

A

A

C

1

IBP Example – Iteration 31R

A P(A)

1 1

… 0

A C P(C|A)

1 2 1

3 2 1

… … 0

A B P(B|A)

1 3 1

… … 0

B C F P(F|B,C)

3 2 1 1

… … … 0

A B D P(D|A,B)

1 3 2 1

… … … 0

D F G P(G|D,F)

2 1 3 1

… … … 0

IBP Example – Iteration 4

2R

4R

3R

5R

6R

A

AB AC

ABD BCF

DFG

B

4 5

3

6

2

B

D F

A

A

A

C

1

1R

A P(A)

1 1

… 0

A C P(C|A)

1 2 1

… … 0

A B P(B|A)

1 3 1

… … 0

B C F P(F|B,C)

3 2 1 1

… … … 0

A B D P(D|A,B)

1 3 2 1

… … … 0

D F G P(G|D,F)

2 1 3 1

… … … 0

A B C D F G Belief1 3 2 2 1 3 1… … … … … … 0

IBP Example – Iteration 5

2R

4R

3R

5R

6R

A

AB AC

ABD BCF

DFG

B

4 5

3

6

2

B

D F

A

A

A

C

1

1R

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

det

erm

inism

?

YES

NO

Networks with Determinism: Coding

N=200, 1000 instances, w*=15

Nets w/o Determinism: bn2ow* = 24

w* = 27

w* = 26

Nets with Determinism: LinkagePe

rcen

tage

Abs

olut

e Er

ror

pedigree1, w* = 21Exact Histogram IJGP Histogram Recall Abs. Error Precision Abs. Error

Perc

enta

ge

Abs

olut

e Er

ror

pedigree37, w* = 30

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

X

Y

Nodes with extreme supportObserved variables with xtreme priors or xtreme

support can nearly-cut information flow:

0

0.00005

0.0001

0.00015

0.0002

0.00001 0.01 0.2 0.5 0.8 0.99 0.99999prior

RootB1C1B2C2Sink

D

B1

ED

A

B2

EBEC

EB EC

C1

C2

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.