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Transcript of Probabilistic Inference Lecture 1 M. Pawan Kumar [email protected] Slides available online
![Page 1: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/1.jpg)
Probabilistic InferenceLecture 1
M. Pawan Kumar
Slides available online http://cvc.centrale-ponts.fr/personnel/pawan/
![Page 2: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/2.jpg)
About the Course
• 7 lectures + 1 exam
• Probabilistic Models – 1 lecture
• Energy Minimization – 4 lectures
• Computing Marginals – 2 lectures
• Related Courses• Probabilistic Graphical Models (MVA)• Structured Prediction
![Page 3: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/3.jpg)
Instructor
• Assistant Professor (2012 – Present)
• Center for Visual Computing• 12 Full-time Faculty Members• 2 Associate Faculty Members
• Research Interests• Probabilistic Models• Machine Learning• Computer Vision• Medical Image Analysis
![Page 4: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/4.jpg)
Students
• Third year at ECP
• Specializing in Machine Learning and Vision
• Prerequisites• Probability Theory• Continuous Optimization• Discrete Optimization
![Page 5: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/5.jpg)
Outline
• Probabilistic Models
• Conversions
• Exponential Family
• Inference
Example (on board) !!
![Page 6: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/6.jpg)
Outline
• Probabilistic Models• Markov Random Fields (MRF)• Bayesian Networks• Factor Graphs
• Conversions
• Exponential Family
• Inference
![Page 7: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/7.jpg)
MRF
UnobservedRandomVariables
Edges define a neighborhood over random variables
Neighbors
![Page 8: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/8.jpg)
MRF
V1 V2 V3
V4 V5 V6
V7 V8 V9
Variable Va takes a value or a label va from a set L
V = v is called a labeling Discrete, Finite
= {l1, l2,…, lh}
![Page 9: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/9.jpg)
MRF
V1 V2 V3
V4 V5 V6
V7 V8 V9
MRF assumes the Markovian property for P(v)
![Page 10: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/10.jpg)
MRF
V1 V2 V3
V4 V5 V6
V7 V8 V9
Va is conditionally independent of Vb given Va’s neighbors
Hammersley-Clifford Theorem
![Page 11: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/11.jpg)
MRF
V1 V2 V3
V4 V5 V6
V7 V8 V9
Probability P(v) can be decomposed into clique potentials
Potentialψ12(v1,v2)
Potentialψ56(v5,v6)
![Page 12: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/12.jpg)
MRF
V1
d1
V2
d2
V3
d3
V4
d4
V5
d5
V6
d6
V7
d7
V8
d8
V9
d9
Probability P(v) proportional to Π(a,b) ψab(va,vb)
Potentialψ1(v1,d1)
Probability P(d|v) proportional to Πa ψa (va,da)
ObservedData
![Page 13: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/13.jpg)
MRF
V1
d1
V2
d2
V3
d3
V4
d4
V5
d5
V6
d6
V7
d7
V8
d8
V9
d9
Probability P(v,d) =Πa ψa(va,da) Π(a,b) ψab(va,vb)
Z
Z is known as the partition function
![Page 14: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/14.jpg)
MRF
V1
d1
V2
d2
V3
d3
V4
d4
V5
d5
V6
d6
V7
d7
V8
d8
V9
d9
High-orderPotential
ψ4578(v4,v5,v7,v8)
![Page 15: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/15.jpg)
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
![Page 16: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/16.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
![Page 17: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/17.jpg)
Conditional Random Fields (CRF)
V1
d1
V2
d2
V3
d3
V4
d4
V5
d5
V6
d6
V7
d7
V8
d8
V9
d9
CRF assumes the Markovian property for P(v|d)
Hammersley-Clifford Theorem
![Page 18: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/18.jpg)
CRF
V1
d1
V2
d2
V3
d3
V4
d4
V5
d5
V6
d6
V7
d7
V8
d8
V9
d9
Probability P(v|d) proportional to Πa ψa(va;d) Π(a,b) ψab(va,vb;d)
Clique potentials that depend on the data
![Page 19: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/19.jpg)
CRF
V1
d1
V2
d2
V3
d3
V4
d4
V5
d5
V6
d6
V7
d7
V8
d8
V9
d9
Probability P(v|d) =Πa ψa (va;d) Π(a,b) ψab(va,vb;d)
Z
Z is known as the partition function
![Page 20: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/20.jpg)
MRF and CRF
V1 V2 V3
V4 V5 V6
V7 V8 V9
Probability P(v) =Πa ψa(va) Π(a,b) ψab(va,vb)
Z
![Page 21: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/21.jpg)
Outline
• Probabilistic Models• Markov Random Fields (MRF)• Bayesian Networks• Factor Graphs
• Conversions
• Exponential Family
• Inference
![Page 22: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/22.jpg)
Bayesian Networks
V1
V2 V3
V4 V5 V6
V7 V8
Directed Acyclic Graph (DAG) – no directed loops
Ignoring directionality of edges, a DAG can have loops
![Page 23: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/23.jpg)
Bayesian Networks
V1
V2 V3
V4 V5 V6
V7 V8
Bayesian Network concisely represents the probability P(v)
![Page 24: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/24.jpg)
Bayesian Networks
V1
V2 V3
V4 V5 V6
V7 V8
Probability P(v) = Πa P(va|Parents(va))
P(v1)P(v2|v1)P(v3|v1)P(v4|v2)P(v5|v2,v3)P(v6|v3)P(v7|v4,v5)P(v8|v5,v6)
![Page 25: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/25.jpg)
Bayesian Networks
Courtesy Kevin Murphy
![Page 26: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/26.jpg)
Bayesian Networks
V1
V2 V3
V4 V5 V6
V7 V8
Va is conditionally independent of its ancestors given its parents
![Page 27: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/27.jpg)
Bayesian Networks
Conditional independence of A and B given C
Courtesy Kevin Murphy
![Page 28: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/28.jpg)
Outline
• Probabilistic Models• Markov Random Fields (MRF)• Bayesian Networks• Factor Graphs
• Conversions
• Exponential Family
• Inference
![Page 29: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/29.jpg)
Factor Graphs
V1 V2a V3b
c
V4 V5f V6g
d e
Two types of nodes: variable nodes and factor nodes
Bipartite graph between the two types of nodes
![Page 30: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/30.jpg)
Factor Graphs
V1 V2a V3b
c
V4 V5f V6g
d e
Factor graphs concisely represents the probability P(v)
ψa(v1,v2)
![Page 31: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/31.jpg)
Factor Graphs
V1 V2a V3b
c
V4 V5f V6g
d e
Factor graphs concisely represents the probability P(v)
ψa({v}a)
![Page 32: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/32.jpg)
Factor Graphs
V1 V2a V3b
c
V4 V5f V6g
d e
Factor graphs concisely represents the probability P(v)
ψb(v2,v3)
![Page 33: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/33.jpg)
Factor Graphs
V1 V2a V3b
c
V4 V5f V6g
d e
Factor graphs concisely represents the probability P(v)
ψb({v}b)
![Page 34: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/34.jpg)
Factor Graphs
V1 V2a V3b
c
V4 V5f V6g
d e
ψb({v}b)
Probability P(v) =Πa ψa({v}a)
Z
Z is known as the partition function
![Page 35: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/35.jpg)
Outline
• Probabilistic Models
• Conversions
• Exponential Family
• Inference
![Page 36: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/36.jpg)
MRF to Factor Graphs
![Page 37: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/37.jpg)
Bayesian Networks to Factor Graphs
![Page 38: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/38.jpg)
Factor Graphs to MRF
![Page 39: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/39.jpg)
Outline
• Probabilistic Models
• Conversions
• Exponential Family
• Inference
![Page 40: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/40.jpg)
Motivation
Random Variable V Label set L = {l1, l2,…, lh}
Samples V1, V2, …, Vm that are i.i.d.
Functions ϕα: L Reals
Empirical expectations: μα = (Σi ϕα(Vi))/m
Expectation wrt distribution P: EP[ϕα(V)] = Σi ϕα(li)P(li)
Given empirical expectations, find compatible distribution
Underdetermined problem
α indexes a set of functions
![Page 41: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/41.jpg)
Maximum Entropy Principle
max Entropy of the distribution
s.t. Distribution is compatible
![Page 42: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/42.jpg)
Maximum Entropy Principle
max -Σi P(li)log(P(li))
s.t. Distribution is compatible
![Page 43: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/43.jpg)
Maximum Entropy Principle
max -Σi P(li)log(P(li))
s.t. Σi ϕα(li)P(li) = μα for all α
Σi P(li) = 1
P(v) proportional to exp(-Σα θαϕα(v))
![Page 44: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/44.jpg)
Exponential Family
Random Variable V = {V1, V2, …,Vn} Label set L = {l1, l2,…, lh}
Labeling V = v, va L for all a {1, 2,…, n}
Functions ϕα: Ln Reals α indexes a set of functions
P(v) = exp{-Σα θαΦα(v) - A(θ)}
SufficientStatistics
Parameters NormalizationConstant
![Page 45: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/45.jpg)
Minimal Representation
P(v) = exp{-Σα θαΦα(v) - A(θ)}
SufficientStatistics
Parameters NormalizationConstant
No non-zero c such that Σα cαΦα(v) = Constant
![Page 46: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/46.jpg)
Ising Model
P(v) = exp{-Σα θαΦα(v) - A(θ)}
Random Variable V = {V1, V2, …,Vn} Label set L = {l1, l2}
![Page 47: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/47.jpg)
Ising Model
P(v) = exp{-Σα θαΦα(v) - A(θ)}
Random Variable V = {V1, V2, …,Vn} Label set L = {-1, +1}
Neighborhood over variables specified by edges E
Sufficient Statistics Parameters
va θafor all Va V
vavb θab for all (Va,Vb) E
![Page 48: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/48.jpg)
Ising Model
P(v) = exp{-Σa θava -Σa,b θabvavb- A(θ)}
Random Variable V = {V1, V2, …,Vn} Label set L = {-1, +1}
Neighborhood over variables specified by edges E
Sufficient Statistics Parameters
va θafor all Va V
vavb θab for all (Va,Vb) E
![Page 49: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/49.jpg)
Interactive Binary Segmentation
![Page 50: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/50.jpg)
Interactive Binary Segmentation
Foreground histogram of RGB values FG
Background histogram of RGB values BG
‘+1’ indicates foreground and ‘-1’ indicates background
![Page 51: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/51.jpg)
Interactive Binary Segmentation
More likely to be foreground than background
![Page 52: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/52.jpg)
Interactive Binary Segmentation
More likely to be background than foreground
θa proportional to -log(FG(da)) + log(BG(da))
![Page 53: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/53.jpg)
Interactive Binary Segmentation
More likely to belong to same label
![Page 54: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/54.jpg)
Interactive Binary Segmentation
Less likely to belong to same label
θab proportional to -exp(-(da-db)2)
![Page 55: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/55.jpg)
Rest of lecture 1 ….
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Exponential Family
P(v) = exp{-Σα θαΦα(v) - A(θ)}
SufficientStatistics
Parameters Log-PartitionFunction
Random Variables V = {V1,V2,…,Vn}
Labeling V = vva L = {l1,l2,…,lh}
Random Variable Va takes a value or label va
![Page 57: Probabilistic Inference Lecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online](https://reader036.fdocuments.us/reader036/viewer/2022062300/56649d005503460f949d2e85/html5/thumbnails/57.jpg)
Overcomplete Representation
P(v) = exp{-Σα θαΦα(v) - A(θ)}
SufficientStatistics
Parameters Log-PartitionFunction
There exists a non-zero c such that Σα cαΦα(v) = Constant
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Ising Model
P(v) = exp{-Σα θαΦα(v) - A(θ)}
Random Variable V = {V1, V2, …,Vn} Label set L = {l1, l2}
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Ising Model
P(v) = exp{-Σα θαΦα(v) - A(θ)}
Random Variable V = {V1, V2, …,Vn} Label set L = {0, 1}
Neighborhood over variables specified by edges E
Sufficient Statistics Parameters
Ia;i(va) θa;ifor all Va V, li L
θab;ik for all (Va,Vb) E, li, lk L
Iab;ik(va,vb)
Ia;i(va): indicator for va = li Iab;ik(va,vb): indicator for va = li, vb = lk
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Ising Model
P(v) = exp{-Σa Σi θa;iIa;i(va) -Σa,b Σi,k θab;ikIab;ik(va,vb) - A(θ)}
Random Variable V = {V1, V2, …,Vn} Label set L = {0, 1}
Neighborhood over variables specified by edges E
Sufficient Statistics Parameters
Ia;i(va) θa;ifor all Va V, li L
θab;ik for all (Va,Vb) E, li, lk L
Iab;ik(va,vb)
Ia;i(va): indicator for va = li Iab;ik(va,vb): indicator for va = li, vb = lk
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Interactive Binary Segmentation
Foreground histogram of RGB values FG
Background histogram of RGB values BG
‘1’ indicates foreground and ‘0’ indicates background
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Interactive Binary Segmentation
More likely to be foreground than background
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Interactive Binary Segmentation
More likely to be background than foreground
θa;0 proportional to -log(BG(da))
θa;1 proportional to -log(FG(da))
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Interactive Binary Segmentation
More likely to belong to same label
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Interactive Binary Segmentation
Less likely to belong to same label
θab;ik proportional to exp(-(da-db)2) if i ≠ k
θab;ik = 0 if i = k
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Metric Labeling
P(v) = exp{-Σα θαΦα(v) - A(θ)}
Random Variable V = {V1, V2, …,Vn} Label set L = {l1, l2, …, lh}
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Metric Labeling
P(v) = exp{-Σα θαΦα(v) - A(θ)}
Random Variable V = {V1, V2, …,Vn}
Neighborhood over variables specified by edges E
Sufficient Statistics Parameters
Ia;i(va) θa;ifor all Va V, li L
θab;ik for all (Va,Vb) E, li, lk L
Iab;ik(va,vb)
θab;ik is a metric distance function over labels
Label set L = {0, …, h-1}
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Metric Labeling
P(v) = exp{-Σa Σi θa;iIa;i(va) -Σa,b Σi,k θab;ikIab;ik(va,vb) - A(θ)}
Random Variable V = {V1, V2, …,Vn}
Neighborhood over variables specified by edges E
Sufficient Statistics Parameters
Ia;i(va) θa;ifor all Va V, li L
θab;ik for all (Va,Vb) E, li, lk L
Iab;ik(va,vb)
θab;ik is a metric distance function over labels
Label set L = {0, …, h-1}
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Stereo Correspondence
Disparity Map
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Stereo Correspondence
L = {disparities}
Pixel (xa,ya) in leftcorresponds to
pixel (xa+va,ya) in right
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Stereo Correspondence
L = {disparities}
θa;i is proportional tothe difference in RGB values
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Stereo Correspondence
L = {disparities}
θab;ik = wab d(i,k)
wab proportional to exp(-(da-db)2)
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Pairwise MRF
Random Variable V = {V1, V2, …,Vn}
Neighborhood over variables specified by edges E
Label set L = {l1, l2, …, lh}
P(v) = exp{-Σα θαΦα(v) - A(θ)}
Sufficient Statistics Parameters
Ia;i(va) θa;ifor all Va V, li L
θab;ik for all (Va,Vb) E, li, lk L
Iab;ik(va,vb)
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Pairwise MRF
Random Variable V = {V1, V2, …,Vn}
Neighborhood over variables specified by edges E
Label set L = {l1, l2, …, lh}
P(v) = exp{-Σa Σi θa;iIa;i(va) -Σa,b Σi,k θab;ikIab;ik(va,vb) - A(θ)}
A(θ) : log Z
Probability P(v) =Πa ψa(va) Π(a,b) ψab(va,vb)
Z
ψa(li) : exp(-θa;i) ψa(li,lk) : exp(-θab;ik)
Parameters θ are sometimes also referred to as potentials
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Pairwise MRF
Random Variable V = {V1, V2, …,Vn}
Neighborhood over variables specified by edges E
Label set L = {l1, l2, …, lh}
P(v) = exp{-Σa Σi θa;iIa;i(va) -Σa,b Σi,k θab;ikIab;ik(va,vb) - A(θ)}
Labeling as a function f : {1, 2, … , n} {1, 2, …, h}
Variable Va takes a label lf(a)
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Pairwise MRF
Random Variable V = {V1, V2, …,Vn}
Neighborhood over variables specified by edges E
Label set L = {l1, l2, …, lh}
P(f) = exp{-Σa θa;f(a) -Σa,b θab;f(a)f(b) - A(θ)}
Labeling as a function f : {1, 2, … , n} {1, 2, …, h}
Variable Va takes a label lf(a)
Energy Q(f) = Σa θa;f(a) + Σa,b θab;f(a)f(b)
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Pairwise MRF
Random Variable V = {V1, V2, …,Vn}
Neighborhood over variables specified by edges E
Label set L = {l1, l2, …, lh}
P(f) = exp{-Q(f) - A(θ)}
Labeling as a function f : {1, 2, … , n} {1, 2, …, h}
Variable Va takes a label lf(a)
Energy Q(f) = Σa θa;f(a) + Σa,b θab;f(a)f(b)
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Outline
• Probabilistic Models
• Conversions
• Exponential Family
• Inference
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Inference
maxv ( P(v) = exp{-Σa Σi θa;iIa;i(va) -Σa,b Σi,k θab;ikIab;ik(va,vb) - A(θ)} )
Maximum a Posteriori (MAP) Estimation
minf ( Q(f) = Σa θa;f(a) + Σa,b θab;f(a)f(b) )
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)
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Next Lecture …
Energy minimization for tree-structured pairwise MRF