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