Probabilistic graphical models
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Transcript of Probabilistic graphical models
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 Branching Application
Directed Markov Chain
(HMM)
Bayesian network (BN)
AI
Statistics
Undirected Linear chain conditional random field (CRF)
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
B C
E
G
D F
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
TATAAATATAATTATAAATATAAATATAAATATTAATTAAAATAGAAA
1 2 3 4 5 6 T 8 1 6 1 0 1C 0 0 0 0 0 0A 0 7 1 7 8 7G 0 0 1 0 0 0
1
A1
2
A2
3
A3
4
A4
5
A5
6
A6
The nucleotide distributions at different sites are independent !
Mixture of profile model
A1 A2 A3 A4 A5 A6
Z
11
m1 1
2m
2 14
m4 1
5m
5
The nt-distributions at different sites are conditionally independent but marginally dependent !
Tree model
1
A1
2
A2
3
A3
4
A4
5
A5
6
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– Do not have to sum to 1– Represent affinities
D
A
BC
A C 1[A,C]
a0 c0 4
a0 c1 12
a1 c0 2
a1 c1 9
A B 2[A,B]
a0 b0 30
a0 b1 5
a1 b0 1
a1 b1 10
C D 3[C,D]
c0 d0 30
c0 d1 5
c1 d0 1
c1 d1 10
B D 4[B,D]
b0 d0 100
b0 d1 1
b1 d0 1
b1 d1 1000
Markov network• Represents joint distribution
– Unnormalized factor
– Partition function
– Probability D
A
BC
],[],[],[],[),,,( 4321 dcdbcabadcbaF
dcba
dcdbcabaZ,,,
4321 ],[],[],[],[
],[],[],[],[1
),,,( 4321 dcdbcabaZ
dcbaP
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
C
A
DB
C
A
DB
Maximal cliques• {A,B}• {B,C}• {C,D}• {A,D}
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,XjH}
– 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
• Solution: inference– Find most likely assignment to Xi variables
X11 X12 X13 X14
X21 X22 X23 X24
X31 X32 X33 X34
Appearance distribution
Contiguity preference