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Probabilistic graphical models
Probabilistic graphical models
• Graphical models are a marriage between
probability theory and graph theory (Michael
Jordan, 1998)
• A compact representation of joint probability
distributions.
• Graphs
– nodes: random variables (probabilistic distribution
over a fixed alphabet)
– edges (arcs), or lack of edges: conditional
independence assumptions
Classification of probabilistic
graphical models
Linear
Directed
Markov
Chain
(HMM)
Undirected Linear chain
conditional
random field
(CRF)
Branching
Application
Bayesian
network
(BN)
AI
Statistics
Markov
network
(MN)
Physics
(Ising)
Image/Vision
Both directed and undirected arcs: chain graphs
Bayesian Network Structure
• Directed acyclic graph G
– Nodes X1,…,Xn represent random variables
• G encodes local Markov assumptions
– Xi is independent of its non-descendants given its
parents
A
D
B
C
E
F
G
Bayesian Network
• Conditional probability
distribution (CPD) at
each node
– T (true), F (false)
• P(C, S, R, W) = P(C) *
P(S|C) * P(R|C,S) *
P(W|C,S,R)  P(C) *
P(S|C) * P(R|C) * P(W|S,R)
• 8 independent parameters
Training Bayesian network:
frequencies
Known: frequencies  Pr(c, s, r, w) for all (c, s, r, w)
Application: Recommendation
Systems
• Given user preferences, suggest recommendations
– Amazon.com
• Input: movie preferences of many users
• Solution: model correlations between movie features
– Users that like comedy, often like drama
– Users that like action, often do not like cartoons
– Users that like Robert Deniro films often like Al
Pacino films
– Given user preferences, can predict probability that
new movies match preferences
Application: modeling DNA motifs
• Profile model: no dependences between
positions
• Markov model: dependence between
adjacent positions
• Bayesian network model: non-local
dependences
A DNA profile
TATAAA
TATAAT
TATAAA
TATAAA
TATAAA
TATTAA
TTAAAA
TAGAAA
1
8
0
0
0
T
C
A
G
1
2
1
0
7
0
3
2
A1
3
6
0
1
1
A2
4
1
0
7
0
4
A3
5
0
0
8
0
5
A4
6
1
0
7
0
6
A5
The nucleotide distributions at different sites are independent !
A6
Mixture of profile model
11
m1 12
A1
m 2
A2
14
A3
m4
A4
15
A5
Z
The nt-distributions at different sites are conditionally
independent but marginally dependent !
m5
A6
Tree model
1
3
2
A1
A2
4
A3
5
A4
6
A5
A6
The nt-distributions at different sites are pairwisely dependent !
Undirected graphical models (e.g.
Markov network)
• Useful when edge directionality cannot be
assigned
• Simpler interpretation of structure
– Simpler inference
– Simpler independency structure
• Harder to learn
Markov network
• Nodes correspond to random variables
• Local factor models are attached to sets of nodes
– Factor elements are positive
A
– Do not have to sum to 1 A C 1[A,C]
a0
c0
4
a0
– Represent affinities
B
2[A,B]
b0
30
a0
c1
12
a0
b1
5
a1
c0
2
a1
b0
1
a1
c1
9
a1
b1
10
A
C
C
D
3[C,D]
c0
d0
30
c0
d1
c1
c1
B
B
D
4[B,D]
b0
d0
100
5
b0
d1
1
d0
1
b1
d0
1
d1
10
b1
d1
1000
D
Markov network
• Represents joint distribution
– Unnormalized factor
F (a, b, c, d )   1[a, b] 2 [a, c] 3[b, d ] 4 [c, d ]
– Partition function
Z
  [a, b]
1
2
[a, c] 3 [b, d ] 4 [c, d ]
a ,b , c , d
– Probability
1
P(a, b, c, d )   1[a, b] 2 [a, c] 3[b, d ] 4 [c, d ]
Z
A
C
B
D
Markov Network Factors
• A factor is a function from value
assignments of a set of random variables
D to real positive numbers
– The set of variables D is the scope of the
factor
• Factors generalize the notion of CPDs
– Every CPD is a factor (with additional
constraints)
Markov Network Factors
A
A
B
D
C
Maximal cliques
• {A,B}
• {B,C}
• {C,D}
• {A,D}
B
D
C
Maximal cliques
• {A,B,C}
• {A,C,D}
Pairwise Markov networks
• A pairwise Markov network over a graph H has:
– A set of node potentials {[Xi]:i=1,...n}
– A set of edge potentials {[Xi,Xj]: Xi,XjH}
– Example: Grid structured Markov network
X11
X12
X13
X14
X21
X22
X23
X24
X31
X32
X33
X34
Application: Image analysis
• The image segmentation problem
– Task: Partition an image into distinct parts of the scene
– Example: separate water, sky, background
Markov Network for Segmentation
• Grid structured Markov network
• Random variable Xi corresponds to pixel i
– Domain is {1,...K}
– Value represents region assignment to pixel i
• Neighboring pixels are connected in the network
• Appearance distribution
– wik – extent to which pixel i “fits” region k (e.g.,
difference from typical pixel for region k)
– Introduce node potential exp(-wik1{Xi=k})
• Edge potentials
– Encodes contiguity preference by edge potential
exp(1{Xi=Xj}) for >0
Markov Network for Segmentation
Appearance
distribution
X11
X12
X13
X14
X21
X22
X23
X24
X31
X32
X33
X34
• Solution: inference
Contiguity
preference
– Find most likely assignment to Xi variables