Discrete Optimization Lecture 5 – Part 2 M. Pawan Kumar [email protected] Slides available online .
Probabilistic Inference Lecture 2 M. Pawan Kumar [email protected] Slides available online
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Transcript of Probabilistic Inference Lecture 2 M. Pawan Kumar [email protected] Slides available online
![Page 1: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/1.jpg)
Probabilistic InferenceLecture 2
M. Pawan Kumar
Slides available online http://cvc.centrale-ponts.fr/personnel/pawan/
![Page 2: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/2.jpg)
Pose Estimation
Courtesy Pedro Felzenszwalb
![Page 3: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/3.jpg)
Pose Estimation
Courtesy Pedro Felzenszwalb
![Page 4: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/4.jpg)
Pose Estimation
Variables are body parts Labels are positions
![Page 5: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/5.jpg)
Pose Estimation
Unary potentials θa;i proportional to fraction of foreground pixels
Variables are body parts Labels are positions
![Page 6: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/6.jpg)
Pose Estimation
Pairwise potentials θab;ik proportional to d2
Head
Torso
Joint location according to ‘head’ part
Joint location according to ‘torso’ partd
![Page 7: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/7.jpg)
Pose Estimation
Pairwise potentials θab;ik proportional to d2
Head
Torso
dHead
Torso
d>
![Page 8: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/8.jpg)
Outline• Problem Formulation
– Energy Function– Energy Minimization– Computing min-marginals
• Reparameterization
• Energy Minimization for Trees
• Loopy Belief Propagation
![Page 9: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/9.jpg)
Energy Function
Va Vb Vc Vd
Label l0
Label l1
Random Variables V = {Va, Vb, ….}
Labels L = {l0, l1, ….}
Labelling f: {a, b, …. } {0,1, …}
![Page 10: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/10.jpg)
Energy Function
Va Vb Vc Vd
Q(f) = ∑a a;f(a)
Unary Potential
2
5
4
2
6
3
3
7Label l0
Label l1
Easy to minimize
Neighbourhood
![Page 11: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/11.jpg)
Energy Function
Va Vb Vc Vd
E : (a,b) E iff Va and Vb are neighbours
E = { (a,b) , (b,c) , (c,d) }
2
5
4
2
6
3
3
7Label l0
Label l1
![Page 12: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/12.jpg)
Energy Function
Va Vb Vc Vd
+∑(a,b) ab;f(a)f(b)
Pairwise Potential
0
1 1
0
0
2
1
1
4 1
0
3
2
5
4
2
6
3
3
7Label l0
Label l1
Q(f) = ∑a a;f(a)
![Page 13: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/13.jpg)
Energy Function
Va Vb Vc Vd
0
1 1
0
0
2
1
1
4 1
0
3
Parameter
2
5
4
2
6
3
3
7Label l0
Label l1
+∑(a,b) ab;f(a)f(b)Q(f; ) = ∑a a;f(a)
![Page 14: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/14.jpg)
Outline• Problem Formulation
– Energy Function– Energy Minimization– Computing min-marginals
• Reparameterization
• Energy Minimization for Trees
• Loopy Belief Propagation
![Page 15: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/15.jpg)
Energy Minimization
Va Vb Vc Vd
2
5
4
2
6
3
3
7
0
1 1
0
0
2
1
1
4 1
0
3
Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)
Label l0
Label l1
![Page 16: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/16.jpg)
Energy Minimization
Va Vb Vc Vd
2
5
4
2
6
3
3
7
0
1 1
0
0
2
1
1
4 1
0
3
Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)
2 + 1 + 2 + 1 + 3 + 1 + 3 = 13
Label l0
Label l1
![Page 17: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/17.jpg)
Energy Minimization
Va Vb Vc Vd
2
5
4
2
6
3
3
7
0
1 1
0
0
2
1
1
4 1
0
3
Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)
Label l0
Label l1
![Page 18: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/18.jpg)
Energy Minimization
Va Vb Vc Vd
2
5
4
2
6
3
3
7
0
1 1
0
0
2
1
1
4 1
0
3
Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)
5 + 1 + 4 + 0 + 6 + 4 + 7 = 27
Label l0
Label l1
![Page 19: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/19.jpg)
Energy Minimization
Va Vb Vc Vd
2
5
4
2
6
3
3
7
0
1 1
0
0
2
1
1
4 1
0
3
Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)
f* = arg min Q(f; )
q* = min Q(f; ) = Q(f*; )
Label l0
Label l1
![Page 20: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/20.jpg)
Energy Minimization
f(a) f(b) f(c) f(d) Q(f; )0 0 0 0 180 0 0 1 150 0 1 0 270 0 1 1 200 1 0 0 220 1 0 1 190 1 1 0 270 1 1 1 20
16 possible labellings
f(a) f(b) f(c) f(d) Q(f; )1 0 0 0 16
1 0 0 1 13
1 0 1 0 25
1 0 1 1 18
1 1 0 0 18
1 1 0 1 15
1 1 1 0 23
1 1 1 1 16
f* = {1, 0, 0, 1}q* = 13
![Page 21: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/21.jpg)
Outline• Problem Formulation
– Energy Function– Energy Minimization– Computing min-marginals
• Reparameterization
• Energy Minimization for Trees
• Loopy Belief Propagation
![Page 22: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/22.jpg)
Min-Marginals
Va Vb Vc Vd
2
5
4
2
6
3
3
7
0
1 1
0
0
2
1
1
4 1
0
3
f* = arg min Q(f; ) such that f(a) = i
Min-marginal qa;i
Label l0
Label l1
![Page 23: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/23.jpg)
Min-Marginals16 possible labellings qa;0 = 15f(a) f(b) f(c) f(d) Q(f; )0 0 0 0 18
0 0 0 1 15
0 0 1 0 27
0 0 1 1 20
0 1 0 0 22
0 1 0 1 19
0 1 1 0 27
0 1 1 1 20
f(a) f(b) f(c) f(d) Q(f; )1 0 0 0 16
1 0 0 1 13
1 0 1 0 25
1 0 1 1 18
1 1 0 0 18
1 1 0 1 15
1 1 1 0 23
1 1 1 1 16
![Page 24: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/24.jpg)
Min-Marginals16 possible labellings qa;1 = 13
f(a) f(b) f(c) f(d) Q(f; )1 0 0 0 16
1 0 0 1 13
1 0 1 0 25
1 0 1 1 18
1 1 0 0 18
1 1 0 1 15
1 1 1 0 23
1 1 1 1 16
f(a) f(b) f(c) f(d) Q(f; )0 0 0 0 18
0 0 0 1 15
0 0 1 0 27
0 0 1 1 20
0 1 0 0 22
0 1 0 1 19
0 1 1 0 27
0 1 1 1 20
![Page 25: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/25.jpg)
Min-Marginals and MAP• Minimum min-marginal of any variable = energy of MAP labelling
minf Q(f; ) such that f(a) = i
qa;i mini
mini ( )
Va has to take one label
minf Q(f; )
![Page 26: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/26.jpg)
Summary
Energy Minimization
f* = arg min Q(f; )
Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)
Min-marginals
qa;i = min Q(f; ) s.t. f(a) = i
Energy Function
![Page 27: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/27.jpg)
Outline• Problem Formulation
• Reparameterization
• Energy Minimization for Trees
• Loopy Belief Propagation
![Page 28: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/28.jpg)
Reparameterization
Va Vb
2
5
4
2
0
1 1
0
f(a) f(b) Q(f; )
0 0 7
0 1 10
1 0 5
1 1 6
2 +
2 +
- 2
- 2
Add a constant to all a;i
Subtract that constant from all b;k
![Page 29: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/29.jpg)
Reparameterization
f(a) f(b) Q(f; )
0 0 7 + 2 - 2
0 1 10 + 2 - 2
1 0 5 + 2 - 2
1 1 6 + 2 - 2
Add a constant to all a;i
Subtract that constant from all b;k
Q(f; ’) = Q(f; )
Va Vb
2
5
4
2
0
0
2 +
2 +
- 2
- 2
1 1
![Page 30: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/30.jpg)
Reparameterization
Va Vb
2
5
4
2
0
1 1
0
f(a) f(b) Q(f; )
0 0 7
0 1 10
1 0 5
1 1 6
- 3 + 3
Add a constant to one b;k
Subtract that constant from ab;ik for all ‘i’
- 3
![Page 31: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/31.jpg)
Reparameterization
Va Vb
2
5
4
2
0
1 1
0
f(a) f(b) Q(f; )
0 0 7
0 1 10 - 3 + 3
1 0 5
1 1 6 - 3 + 3
- 3 + 3
- 3
Q(f; ’) = Q(f; )
Add a constant to one b;k
Subtract that constant from ab;ik for all ‘i’
![Page 32: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/32.jpg)
Reparameterization
Va Vb
2
5
4
2
3 1
0
1
2
Va Vb
2
5
4
2
3 1
1
0
1
- 2
- 2
- 2 + 2+ 1
+ 1
+ 1
- 1
Va Vb
2
5
4
2
3 1
2
1
0 - 4 + 4
- 4
- 4
’a;i = a;i ’b;k = b;k
’ab;ik = ab;ik
+ Mab;k
- Mab;k
+ Mba;i
- Mba;i Q(f; ’) = Q(f; )
![Page 33: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/33.jpg)
Reparameterization
Q(f; ’) = Q(f; ), for all f
’ is a reparameterization of , iff
’
’b;k = b;k
’a;i = a;i
’ab;ik = ab;ik
+ Mab;k
- Mab;k
+ Mba;i
- Mba;i
Equivalently Kolmogorov, PAMI, 2006
Va Vb
2
5
4
2
0
0
2 +
2 +
- 2
- 2
1 1
![Page 34: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/34.jpg)
RecapEnergy Minimization
f* = arg min Q(f; )Q(f; ) = ∑a a;f(a) + ∑(a,b) ab;f(a)f(b)
Min-marginals
qa;i = min Q(f; ) s.t. f(a) = i
Q(f; ’) = Q(f; ), for all f ’ Reparameterization
![Page 35: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/35.jpg)
Outline• Problem Formulation
• Reparameterization
• Energy Minimization for Trees
• Loopy Belief Propagation
Pearl, 1988
![Page 36: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/36.jpg)
Belief Propagation
• Belief Propagation is exact for chains
• Some problems are easy
• Exact MAP for trees
• Clever Reparameterization
![Page 37: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/37.jpg)
Two Variables
Va Vb
2
5 2
1
0
Va Vb
2
5
40
1
Choose the right constant ’b;k = qb;k
Add a constant to one b;k
Subtract that constant from ab;ik for all ‘i’
![Page 38: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/38.jpg)
Va Vb
2
5 2
1
0
Va Vb
2
5
40
1
Choose the right constant ’b;k = qb;k
a;0 + ab;00 = 5 + 0
a;1 + ab;10 = 2 + 1minMab;0 =
Two Variables
![Page 39: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/39.jpg)
Va Vb
2
5 5
-2
-3
Va Vb
2
5
40
1
Choose the right constant ’b;k = qb;k
Two Variables
![Page 40: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/40.jpg)
Va Vb
2
5 5
-2
-3
Va Vb
2
5
40
1
Choose the right constant ’b;k = qb;k
f(a) = 1
’b;0 = qb;0
Two Variables
Potentials along the red path add up to 0
![Page 41: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/41.jpg)
Va Vb
2
5 5
-2
-3
Va Vb
2
5
40
1
Choose the right constant ’b;k = qb;k
a;0 + ab;01 = 5 + 1
a;1 + ab;11 = 2 + 0minMab;1 =
Two Variables
![Page 42: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/42.jpg)
Va Vb
2
5 5
-2
-3
Va Vb
2
5
6-2
-1
Choose the right constant ’b;k = qb;k
f(a) = 1
’b;0 = qb;0
f(a) = 1
’b;1 = qb;1
Minimum of min-marginals = MAP estimate
Two Variables
![Page 43: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/43.jpg)
Va Vb
2
5 5
-2
-3
Va Vb
2
5
6-2
-1
Choose the right constant ’b;k = qb;k
f(a) = 1
’b;0 = qb;0
f(a) = 1
’b;1 = qb;1
f*(b) = 0 f*(a) = 1
Two Variables
![Page 44: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/44.jpg)
Va Vb
2
5 5
-2
-3
Va Vb
2
5
6-2
-1
Choose the right constant ’b;k = qb;k
f(a) = 1
’b;0 = qb;0
f(a) = 1
’b;1 = qb;1
We get all the min-marginals of Vb
Two Variables
![Page 45: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/45.jpg)
RecapWe only need to know two sets of equations
General form of Reparameterization
’a;i = a;i
’ab;ik = ab;ik
+ Mab;k
- Mab;k
+ Mba;i
- Mba;i
’b;k = b;k
Reparameterization of (a,b) in Belief Propagation
Mab;k = mini { a;i + ab;ik }
Mba;i = 0
![Page 46: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/46.jpg)
Three Variables
Va Vb
2
5 2
1
0
Vc
4 60
1
0
1
3
2 3
Reparameterize the edge (a,b) as before
l0
l1
![Page 47: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/47.jpg)
Va Vb
2
5 5-3
Vc
6 60
1
-2
3
Reparameterize the edge (a,b) as before
f(a) = 1
f(a) = 1
-2 -1 2 3
Three Variables
l0
l1
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Va Vb
2
5 5-3
Vc
6 60
1
-2
3
Reparameterize the edge (a,b) as before
f(a) = 1
f(a) = 1
Potentials along the red path add up to 0
-2 -1 2 3
Three Variables
l0
l1
![Page 49: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/49.jpg)
Va Vb
2
5 5-3
Vc
6 60
1
-2
3
Reparameterize the edge (b,c) as before
f(a) = 1
f(a) = 1
Potentials along the red path add up to 0
-2 -1 2 3
Three Variables
l0
l1
![Page 50: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/50.jpg)
Va Vb
2
5 5-3
Vc
6 12-6
-5
-2
9
Reparameterize the edge (b,c) as before
f(a) = 1
f(a) = 1
Potentials along the red path add up to 0
f(b) = 1
f(b) = 0
-2 -1 -4 -3
Three Variables
l0
l1
![Page 51: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/51.jpg)
Va Vb
2
5 5-3
Vc
6 12-6
-5
-2
9
Reparameterize the edge (b,c) as before
f(a) = 1
f(a) = 1
Potentials along the red path add up to 0
f(b) = 1
f(b) = 0
qc;0
qc;1-2 -1 -4 -3
Three Variables
l0
l1
![Page 52: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/52.jpg)
Va Vb
2
5 5-3
Vc
6 12-6
-5
-2
9
f(a) = 1
f(a) = 1
f(b) = 1
f(b) = 0
qc;0
qc;1
f*(c) = 0 f*(b) = 0 f*(a) = 1
Generalizes to any length chain
-2 -1 -4 -3
Three Variables
l0
l1
![Page 53: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/53.jpg)
Va Vb
2
5 5-3
Vc
6 12-6
-5
-2
9
f(a) = 1
f(a) = 1
f(b) = 1
f(b) = 0
qc;0
qc;1
f*(c) = 0 f*(b) = 0 f*(a) = 1
Dynamic Programming
-2 -1 -4 -3
Three Variables
l0
l1
![Page 54: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/54.jpg)
Dynamic Programming
3 variables 2 variables + book-keeping
n variables (n-1) variables + book-keeping
Start from left, go to right
Reparameterize current edge (a,b)
Mab;k = mini { a;i + ab;ik }
’ab;ik = ab;ik+ Mab;k - Mab;k’b;k = b;k
Repeat
![Page 55: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/55.jpg)
Dynamic Programming
Start from left, go to right
Reparameterize current edge (a,b)
Mab;k = mini { a;i + ab;ik }
Repeat
Messages Message Passing
Why stop at dynamic programming?
’ab;ik = ab;ik+ Mab;k - Mab;k’b;k = b;k
![Page 56: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/56.jpg)
Va Vb
2
5 5-3
Vc
6 12-6
-5
-2
9
Reparameterize the edge (c,b) as before
-2 -1 -4 -3
Three Variables
l0
l1
![Page 57: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/57.jpg)
Va Vb
2
5 9-3
Vc
11 12-11
-9
-2
9
Reparameterize the edge (c,b) as before
-2 -1 -9 -7
’b;i = qb;i
Three Variables
l0
l1
![Page 58: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/58.jpg)
Va Vb
2
5 9-3
Vc
11 12-11
-9
-2
9
Reparameterize the edge (b,a) as before
-2 -1 -9 -7
Three Variables
l0
l1
![Page 59: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/59.jpg)
Va Vb
9
11 9-9
Vc
11 12-11
-9
-9
9
Reparameterize the edge (b,a) as before
-9 -7 -9 -7
’a;i = qa;i
Three Variables
l0
l1
![Page 60: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/60.jpg)
Va Vb
9
11 9-9
Vc
11 12-11
-9
-9
9
Forward Pass Backward Pass
-9 -7 -9 -7
All min-marginals are computed
Three Variables
l0
l1
![Page 61: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/61.jpg)
Belief Propagation on Chains
Start from left, go to right
Reparameterize current edge (a,b)
Mab;k = mini { a;i + ab;ik }
’ab;ik = ab;ik+ Mab;k - Mab;k’b;k = b;k
Repeat till the end of the chain
Start from right, go to left
Repeat till the end of the chain
![Page 62: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/62.jpg)
Belief Propagation on Chains
• A way of computing reparam constants
• Generalizes to chains of any length
• Forward Pass - Start to End• MAP estimate• Min-marginals of final variable
• Backward Pass - End to start• All other min-marginals
![Page 63: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/63.jpg)
Computational Complexity
• Each constant takes O(|L|)
• Number of constants - O(|E||L|)
O(|E||L|2)
• Memory required ?
O(|E||L|)
![Page 64: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/64.jpg)
Belief Propagation on Trees
Vb
Va
Forward Pass: Leaf Root
All min-marginals are computed
Backward Pass: Root Leaf
Vc
Vd Ve Vg Vh
![Page 65: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/65.jpg)
Outline• Problem Formulation
• Reparameterization
• Energy Minimization for Trees
• Loopy Belief Propagation
Pearl, 1988; Murphy et al., 1999
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Belief Propagation on Cycles
Va Vb
Vd Vc
Where do we start? Arbitrarily
a;0
a;1
b;0
b;1
d;0
d;1
c;0
c;1
Reparameterize (a,b)
![Page 67: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/67.jpg)
Belief Propagation on Cycles
Va Vb
Vd Vc
a;0
a;1
’b;0
’b;1
d;0
d;1
c;0
c;1
Potentials along the red path add up to 0
![Page 68: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/68.jpg)
Belief Propagation on Cycles
Va Vb
Vd Vc
a;0
a;1
’b;0
’b;1
d;0
d;1
’c;0
’c;1
Potentials along the red path add up to 0
![Page 69: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/69.jpg)
Belief Propagation on Cycles
Va Vb
Vd Vc
a;0
a;1
’b;0
’b;1
’d;0
’d;1
’c;0
’c;1
Potentials along the red path add up to 0
![Page 70: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/70.jpg)
Belief Propagation on Cycles
Va Vb
Vd Vc
’a;0
’a;1
’b;0
’b;1
’d;0
’d;1
’c;0
’c;1
Potentials along the red path add up to 0
![Page 71: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/71.jpg)
Belief Propagation on Cycles
Va Vb
Vd Vc
’a;0
’a;1
’b;0
’b;1
’d;0
’d;1
’c;0
’c;1
Potentials along the red path add up to 0
- a;0
- a;1
![Page 72: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/72.jpg)
Belief Propagation on Cycles
Va Vb
Vd Vc
a;0
a;1
b;0
b;1
d;0
d;1
c;0
c;1
Any suggestions? Fix Va to label l0
![Page 73: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/73.jpg)
Belief Propagation on Cycles
Va Vb
Vd Vc
Any suggestions? Fix Va to label l0
a;0 b;0
b;1
d;0
d;1
c;0
c;1
Equivalent to a tree-structured problem
![Page 74: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/74.jpg)
Belief Propagation on Cycles
Va Vb
Vd Vc
a;1
b;0
b;1
d;0
d;1
c;0
c;1
Any suggestions? Fix Va to label l1
Equivalent to a tree-structured problem
![Page 75: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/75.jpg)
Belief Propagation on Cycles
Choose the minimum energy solution
Va Vb
Vd Vc
a;0
a;1
b;0
b;1
d;0
d;1
c;0
c;1
This approach quickly becomes infeasible
![Page 76: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/76.jpg)
Loopy Belief Propagation
V1 V2 V3
V4 V5 V6
V7 V8 V9
Keep reparameterizing edges in some order
Hope for convergence and a good solution
![Page 77: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/77.jpg)
Belief Propagation
• Generalizes to any arbitrary random field
• Complexity per iteration ?
O(|E||L|2)
• Memory required ?
O(|E||L|)
![Page 78: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/78.jpg)
Theoretical Properties of BP
Exact for Trees Pearl, 1988
What about any general random field?
Run BP. Assume it converges.
![Page 79: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/79.jpg)
Theoretical Properties of BP
Exact for Trees Pearl, 1988
What about any general random field?
Choose variables in a tree. Change their labels.Value of energy does not decrease
![Page 80: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/80.jpg)
Theoretical Properties of BP
Exact for Trees Pearl, 1988
What about any general random field?
Choose variables in a cycle. Change their labels.Value of energy does not decrease
![Page 81: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/81.jpg)
Theoretical Properties of BP
Exact for Trees Pearl, 1988
What about any general random field?
For cycles, if BP converges then exact MAPWeiss and Freeman, 2001
![Page 82: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/82.jpg)
Speed-Ups for Special Cases
ab;ik = 0, if i = k
= C, otherwise.
Mab;k = mini { a;i + ab;ik }
Felzenszwalb and Huttenlocher, 2004
![Page 83: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/83.jpg)
Speed-Ups for Special Cases
ab;ik = wab|i-k|
Mab;k = mini { a;i + ab;ik }
Felzenszwalb and Huttenlocher, 2004
![Page 84: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/84.jpg)
Speed-Ups for Special Cases
ab;ik = min{wab|i-k|, C}
Mab;k = mini { a;i + ab;ik }
Felzenszwalb and Huttenlocher, 2004
![Page 85: Probabilistic Inference Lecture 2 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062519/5697c0251a28abf838cd55f2/html5/thumbnails/85.jpg)
Speed-Ups for Special Cases
ab;ik = min{wab(i-k)2, C}
Mab;k = mini { a;i + ab;ik }
Felzenszwalb and Huttenlocher, 2004