probabilistic inference lecture 6 – part 2
DESCRIPTION
Probabilistic Inference Lecture 6 – Part 2. M. Pawan Kumar [email protected]. Slides available online http:// cvc.centrale-ponts.fr /personnel/ pawan /. MRF. d 1. d 2. d 3. V 1. V 2. V 3. d 4. d 5. d 6. V 4. V 5. V 6. d 7. d 8. d 9. V 7. V 8. V 9. - PowerPoint PPT PresentationTRANSCRIPT
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Probabilistic InferenceLecture 6 – Part 2
M. Pawan [email protected]
Slides available online http://cvc.centrale-ponts.fr/personnel/pawan/
![Page 2: Probabilistic Inference Lecture 6 – Part 2](https://reader036.vdocument.in/reader036/viewer/2022062310/568161ce550346895dd1bf69/html5/thumbnails/2.jpg)
MRF
V1
d1
V2
d2
V3
d3
V4
d4
V5
d5
V6
d6
V7
d7
V8
d8
V9
d9
A is conditionally independent of B given C if
there is no path from A to B when C is removed
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MRF
V1
d1
V2
d2
V3
d3
V4
d4
V5
d5
V6
d6
V7
d7
V8
d8
V9
d9
Va is conditionally independent of Vb given Va’s neighbors
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Pairwise MRF
V1
d1
V2
d2
V3
d3
V4
d4
V5
d5
V6
d6
V7
d7
V8
d8
V9
d9
Z is known as the partition function
UnaryPotentialψ1(v1,d1)
PairwisePotentialψ56(v5,v6)
Probability P(v,d) =Πa ψa(va,da) Π(a,b) ψab(va,vb)
Z
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Inference
maxv P(v) Maximum a Posteriori (MAP) Estimation
minv Q(v) Energy Minimization
P(va = li) = Σv P(v)δ(va = li)
Computing Marginals
P(va = li, vb = lk) = Σv P(v)δ(va = li)δ(vb = lk)
P(v) = exp(-Q(v))/Z
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Outline
• Belief Propagation on Chains
• Belief Propagation on Trees
• Loopy Belief Propagation
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Overview
Va Vb Vc Vd
Compute the marginal probability for Vd
P(v) = P(va|vb)P(vb|vc)P(vc|vd)P(vd)
Compute (unnormalized) distribution
Ψa(va)Ψab(va,vb)Σva
Function m(vb)
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Overview
Va Vb Vc Vd
Compute the marginal probability for Vd
P(v) = P(va|vb)P(vb|vc)P(vc|vd)P(vd)
Compute (unnormalized) distribution
Ψb(vb)Ψbc(vb,vc)m(vb)Σvb
Function m(vc)
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Overview
Va Vb Vc Vd
Compute the marginal probability for Vd
P(v) = P(va|vb)P(vb|vc)P(vc|vd)P(vd)
Compute (unnormalized) distribution
Ψc(vc)Ψcd(vc,vd)m(vc)Σvc
(Unnormalized) Marginals !!
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Overview
Va Vb Vc Vd
Compute the marginal probability for Vc
P(v) = P(va|vb)P(vb|vc)P(vc|vd)P(vd)
P(v) = P(va|vb)P(vb|vc)P(vd|vc)P(vc)
Several common terms !!
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Overview
Va Vb Vc Vd
Compute the marginal probability for Vb
P(v) = P(va|vb)P(vb|vc)P(vc|vd)P(vd)
P(v) = P(va|vb)P(vb|vc)P(vd|vc)P(vc)
P(v) = P(va|vb)P(vc|vb)P(vd|vc)P(vb)
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Overview
Va Vb Vc Vd
Compute the marginal probability for Va
P(v) = P(va|vb)P(vb|vc)P(vc|vd)P(vd)
P(v) = P(va|vb)P(vb|vc)P(vd|vc)P(vc)
P(v) = P(va|vb)P(vc|vb)P(vd|vc)P(vb)
P(v) = P(vb|va)P(vc|vb)P(vd|vc)P(va)
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Belief Propagation on Chains
Compute exact marginals
Avoids re-computing common terms
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Two Variables
Va Vb
2
5 2
3
1Va Vb
2
5
41
3
Unary Potentials ψa(li)
Pairwise Potentials ψab(li,lk)
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Two Variables
Va Vb
2
5 2
3
1Va Vb
2
5
41
3
Marginal Probability P(vb = lj) = Σi ψa(li)ψb(lj)ψab(li,lj)/Z
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Two Variables
Va Vb
2
5 2
3
1Va Vb
2
5
41
3
Un-normalized
Marginal Probability P’(vb = lj) = Σi ψa(li)ψb(lj)ψab(li,lj)/Z
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Two Variables
Va Vb
2
5 2
3
1Va Vb
2
5
41
3
Un-normalized
Marginal Probability P’(vb = lj) = Σi ψa(li)ψb(lj)ψab(li,lj)
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Two Variables
Va Vb
2
5 2
3
1Va Vb
2
5
41
3
Un-normalized
Marginal Probability P’(vb = lj) = ψb(lj)Σi ψa(li)ψab(li,lj)
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Two Variables
Va Vb
2
5 2
3
1Va Vb
2
5
41
3
2 x 3
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Two Variables
Va Vb
2
5 2
3
1Va Vb
2
5
41
3
2 x 3 + 5 x 1 Mab;0
11
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Two Variables
Va Vb
2
5
41
3
2 x 1
Va Vb
2
5 2
3
1
11
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Two Variables
2 x 1
11
Va Vb
2
5 2
3
1Va Vb
2
5
41
3
+ 5 x 3 Mab;1
17
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Two Variables11
Va Vb
2
5 2
3
1
17
Marginal Probability P’(vb = lj) = ψb(lj)Σi ψa(li)ψab(li,lj)
Va Vb
2
5
41
3
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Two Variables11
Va Vb
2
5 2
3
1
17
Marginal Probability P’(vb = lj) = ψb(lj)Mab;j
Va Vb
2
5
41
3
P’(vb = l0) = 22 P’(vb = l1) = 68
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Two Variables11
Va Vb
2
5 2
3
1
17
Marginal Probability P(vb = lj) = ψb(lj)Mab;j/Z
Va Vb
2
5
41
3
P’(vb = l0) = 22 P’(vb = l1) = 68
Z = Σj P’(vb = lj) = 90
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Two Variables11
Va Vb
2
5 2
3
1
17
Va Vb
2
5
41
3
P(vb = l0) = 0.244… P(vb = l1) = 0.755…
= 90 O(h2)!!
Marginal Probability P(vb = lj) = ψb(lj)Mab;j/Z
Z = Σj P’(vb = lj)
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Two Variables11
Va Vb
2
5 2
3
1
17
Va Vb
2
5
41
3
P(vb = l0) = 0.244… P(vb = l1) = 0.755…
O(h2)!!Same as brute-force
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Three Variables
Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
P’(vc = lk) Σj Σi ψa(li)ψb(lj)ψc(lk)ψab(li,lj)ψbc(lj,lk)
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Three Variables
Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
P’(vc = lk) ψc(lk)Σj Σi ψa(li)ψb(lj)ψab(li,lj)ψbc(lj,lk)
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Three Variables
Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
P’(vc = lk) ψc(lk)Σj ψb(lj)Σi ψa(li)ψab(li,lj)ψbc(lj,lk)
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Three Variables
Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
P’(vc = lk) ψc(lk)Σj ψb(lj)ψbc(lj,lk)Σi ψa(li)ψab(li,lj)
Mab;j
11
17
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Three Variables
Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
P’(vc = lk) ψc(lk)Σj ψb(lj)ψbc(lj,lk)Mab;j
11
17 Mbc;k
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Three Variables
Va Vb
2
5 2
3
1Vc
4 6
2
1
3
3
2
P’(vc = lk) ψc(lk)Σj ψb(lj)ψbc(lj,lk)Mab;j
11
17
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Three Variables
Va Vb
2
5 2
3
1Vc
4 6
2
1
3
3
2
P’(vc = lk) ψc(lk)Σj ψb(lj)ψbc(lj,lk)Mab;j
11
17
4 x 2 x 11
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Three Variables
Va Vb
2
5 2
3
1Vc
4 6
2
1
3
3
2
P’(vc = lk) ψc(lk)Σj ψb(lj)ψbc(lj,lk)Mab;j
11
17
4 x 2 x 11+ 2 x 2 x 17
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Three Variables
Va Vb
2
5 2
3
1Vc
4 6
2
1
3
3
2
P’(vc = lk) ψc(lk)Σj ψb(lj)ψbc(lj,lk)Mab;j
11
17
4 x 2 x 11+ 2 x 2 x 17
156
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Three Variables
P’(vc = lk) ψc(lk)Σj ψb(lj)ψbc(lj,lk)Mab;j
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11 146
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Three Variables
P’(vc = lk) ψc(lk)Mbc;k
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
NOTE: Mbc;k “includes” Mab;j
146
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Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
P(vc = 0) = 0.35
P(vc = 1) = 0.65
Z = 156 x 3 + 146 x 6 = 1344
146
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Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
O(nh2) Better than brute-force
146
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Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
What about P(vb = lj)?
146
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Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
P’(vb = lj) Σk Σi ψa(li)ψb(lj)ψc(lk)ψab(li,lj)ψbc(lj,lk)
146
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Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
P’(vb = lj) ψb(lj)Σk Σi ψa(li)ψc(lk)ψab(li,lj)ψbc(lj,lk)
146
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Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
P’(vb = lj) ψb(lj)Σk ψc(lk)Σi ψa(li)ψab(li,lj)ψbc(lj,lk)
146
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Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
P’(vb = lj) ψb(lj)Σk ψc(lk)ψbc(lj,lk)Σi ψa(li)ψab(li,lj)
Mab;j
146
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Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
P’(vb = lj) ψb(lj)Mab;jΣk ψc(lk)ψbc(lj,lk)
Mcb;j
NOTE: Mcb;j does not “include” Mbc;k
146
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Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
P’(vb = lj) ψb(lj)Mab;jMcb;j
24
12 146
![Page 48: Probabilistic Inference Lecture 6 – Part 2](https://reader036.vdocument.in/reader036/viewer/2022062310/568161ce550346895dd1bf69/html5/thumbnails/48.jpg)
Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
24
12 146
P(vb = 0) = 0.39
P(vb = 1) = 0.61
Z = 11 x 12 x 4 + 17 x 24 x 2 = 1344
![Page 49: Probabilistic Inference Lecture 6 – Part 2](https://reader036.vdocument.in/reader036/viewer/2022062310/568161ce550346895dd1bf69/html5/thumbnails/49.jpg)
Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
24
12 146
O(nh2) Better than brute-force
![Page 50: Probabilistic Inference Lecture 6 – Part 2](https://reader036.vdocument.in/reader036/viewer/2022062310/568161ce550346895dd1bf69/html5/thumbnails/50.jpg)
Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
24
12 146
What about P(va = li)?
![Page 51: Probabilistic Inference Lecture 6 – Part 2](https://reader036.vdocument.in/reader036/viewer/2022062310/568161ce550346895dd1bf69/html5/thumbnails/51.jpg)
Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
24
12 146
P’(va = li) Σj Σk ψa(li)ψb(lj)ψc(lk)ψab(li,lj)ψbc(lj,lk)
![Page 52: Probabilistic Inference Lecture 6 – Part 2](https://reader036.vdocument.in/reader036/viewer/2022062310/568161ce550346895dd1bf69/html5/thumbnails/52.jpg)
Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
24
12 146
P’(va = li) ψa(li)Σj Σk ψb(lj)ψc(lk)ψab(li,lj)ψbc(lj,lk)
![Page 53: Probabilistic Inference Lecture 6 – Part 2](https://reader036.vdocument.in/reader036/viewer/2022062310/568161ce550346895dd1bf69/html5/thumbnails/53.jpg)
Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
24
12 146
P’(va = li) ψa(li)Σj ψb(lj)Σk ψc(lk)ψab(li,lj)ψbc(lj,lk)
![Page 54: Probabilistic Inference Lecture 6 – Part 2](https://reader036.vdocument.in/reader036/viewer/2022062310/568161ce550346895dd1bf69/html5/thumbnails/54.jpg)
Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
24
12 146
P’(va = li) ψa(li)Σj ψb(lj)ψab(li,lj)Σk ψc(lk)ψbc(lj,lk)
Mcb;j
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Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
24
12 146
P’(va = li) ψa(li)Σj ψb(lj)ψab(li,lj)Mcb;j Mba;i
NOTE: Mba;i “includes” Mcb;j
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Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
24
12 146
P’(va = li) ψa(li)Mba;i
192
192
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Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
24
12 146
192
192
P(va = 0) = 0.71
P(vb = 1) = 0.29
Z = 192 x 2 + 192 x 5 = 1344
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Three Variables
17 156Va Vb
2
5 2
3
1Vc
4 61
2
1
3
3
2 3
11
24
12 146
192
192
O(nh2) Better than brute-force
![Page 59: Probabilistic Inference Lecture 6 – Part 2](https://reader036.vdocument.in/reader036/viewer/2022062310/568161ce550346895dd1bf69/html5/thumbnails/59.jpg)
Belief Propagation on Chains
Start from left, go to right
For current edge (a,b), compute
Mab;k = Σiψa(li)ψab(li,lk)Πn≠bMna;i
Repeat till the end of the chain
Start from right, go to left
Mab;k = Σiψa(li)ψab(li,lk)Πn≠bMna;i
Repeat till the end of the chain
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Belief Propagation on Chains
P’(va = li,vb = lj) = ?
Normalize to compute true marginals
P’(va = li) = ?
ψa(li)ψb(lj)ψab(li,lj)Πn≠bMna;iΠn≠aMnb;j
ψa(li)ΠnMna;i
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Outline
• Belief Propagation on Chains
• Belief Propagation on Trees
• Loopy Belief Propagation
Pearl, 1988
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Belief Propagation on Trees
Vc
Vd
Va Vb
ΣkΣj Σi ψa(li)ψb(lj)ψc(lk)ψd(lo)ψac(li,lk)ψbc(lj,lk)ψcd(lk,lo)
P’(vd = lo)
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Belief Propagation on Trees
Vc
Vd
Va Vb
ψd(lo)ΣkΣj Σi ψa(li)ψb(lj)ψc(lk)ψac(li,lk)ψbc(lj,lk)ψcd(lk,lo)
P’(vd = lo)
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Belief Propagation on Trees
Vc
Vd
Va Vb
ψd(lo)Σkψc(lk)Σj Σi ψa(li)ψb(lj)ψac(li,lk)ψbc(lj,lk)ψcd(lk,lo)
P’(vd = lo)
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Belief Propagation on Trees
Vc
Vd
Va Vb
ψd(lo)Σkψc(lk)ψcd(lk,lo)Σj Σi ψa(li)ψb(lj)ψac(li,lk)ψbc(lj,lk)
P’(vd = lo)
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Belief Propagation on Trees
Vc
Vd
Va Vb
ψd(lo)Σkψc(lk)ψcd(lk,lo)Σj ψb(lj)Σi ψa(li)ψac(li,lk)ψbc(lj,lk)
P’(vd = lo)
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Belief Propagation on Trees
Vc
Vd
Va Vb
ψd(lo)Σkψc(lk)ψcd(lk,lo)Σj ψb(lj)ψbc(lj,lk)Σi ψa(li)ψac(li,lk)
P’(vd = lo) Mac;k
Mac;k
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Belief Propagation on Trees
Vc
Vd
Va Vb
ψd(lo)Σkψc(lk)ψcd(lk,lo)Σj ψb(lj)ψbc(lj,lk)Mac;k
P’(vd = lo) Mbc;k
Mac;k Mbc;k
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Belief Propagation on Trees
Vc
Vd
Va Vb
ψd(lo)Σkψc(lk)ψcd(lk,lo)Mbc;kMac;k
P’(vd = lo)
Mac;k Mbc;k
Mcd;o
Mcd;o
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Belief Propagation on Trees
Vc
Vd
Va Vb
ψd(lo)Mcd;o
P’(vd = lo)
Mac;k Mbc;k
Mcd;o
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Belief Propagation on Trees
Vc
Vd
Va Vb
P’(vc = lk)
Mac;k Mbc;k
Mcd;o
Mdc;k
ψc(lk)Mac;kMbc;kMdc;k
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Belief Propagation on Trees
Vc
Vd
Va Vb
P’(vb = lj)
Mac;k Mbc;k
Mcd;o
Mdc;k
Mcb;j
ψb(lj)Mcb;j
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Belief Propagation on Trees
Vc
Vd
Va Vb
P’(va = li)
Mac;k Mbc;k
Mcd;o
Mdc;k
Mcb;jMca;i
ψa(li)Mca;i
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Belief Propagation on Trees
Start from leaf, go towards root
For current edge (a,b), compute
Mab;k = Σiψa(li)ψab(li,lk)Πn≠bMna;i
Repeat till the root is reached
Start from root, go towards leaves
Mab;k = Σiψa(li)ψab(li,lk)Πn≠bMna;i
Repeat till the leafs are reached
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Belief Propagation on Trees
P’(va = li,vb = lj) = ?
Normalize to compute true marginals
P’(va = li) = ?
ψa(li)ψb(lj)ψab(li,lj)Πn≠bMna;iΠn≠aMnb;j
ψa(li)ΠnMna;i
![Page 76: Probabilistic Inference Lecture 6 – Part 2](https://reader036.vdocument.in/reader036/viewer/2022062310/568161ce550346895dd1bf69/html5/thumbnails/76.jpg)
Outline
• Belief Propagation on Chains
• Belief Propagation on Trees
• Loopy Belief Propagation
Pearl, 1988; Murphy et al., 1999
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Loopy Belief Propagation
Initialize all messages to 1
In some order of edges, update messages
Mab;k = Σiψa(li)ψab(li,lk)Πn≠bMna;i
Until Convergence
Rate of changes in messages < threshold
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Loopy Belief Propagation
Va Vb
Vd Vc
Mab
Mbc
Mbc contains Mab
Mcd
Mda
Mcd contains Mbc
Mda contains Mcd
Overcounting!!
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Loopy Belief Propagation
Initialize all messages to 1
In some order of edges, update messages
Mab;k = Σiψa(li)ψab(li,lk)Πn≠bMna;i
Until Convergence
Rate of changes in messages < threshold
Not Guaranteed !!
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Loopy Belief Propagation
B’ab(i,j) =
Normalize to compute beliefs Ba(i), Bab(i,j)
B’a(i) =
ψa(li)ψb(lj)ψab(li,lj)Πn≠bMna;iΠn≠aMnb;j
ψa(li)ΠnMna;i
At convergence Σj Bab(i,j) = Ba(i)