1. Log-Linear Models Local Structure Representation Probabilistic
Graphical
Models
Local Structure
Log-Linear Models

2. Log-Linear Representation
Each feature fj has a scope Dj
Different features can have same scope

3. Representing Table Factors
(X1, X2) =

4. Features for Language
Features: word capitalized, word in atlas or name list, previous word is “Mrs”, next word is “Times”, …
Phrases

5. Ising Model

6. Metric MRFs All Xi take values in label space V
Distance function  : V  V  R
Reflexivity: (v,v) = 0 for all v
Symmetry: (v1,v2) = (v2,v1) for all v1, v2
Triangle inequality: (v1,v2)  (v1,v3) + (v3, v2) for all v1, v2, v3
want Xi and Xj to
take “similar” values Xi Xj

7. Metric MRFs All Xi take values in label space V
Distance function  : V  V  R
want Xi and Xj to
take “similar” values Xi Xj
values of Xi and Xj far in 
lower probability

8. Metric MRF Examples 1 (vk,vl) = vk=vl otherwise (vk,vl) vk-vl1 1
(vk,vl) =
vk=vl
otherwise
(vk,vl)
vk-vl
(vk,vl)
vk-vl

9. Metric MRF: Segmentation1 1
(vk,vl) =
vk=vl
otherwise

10. Metric MRF: Denoising (vk,vl) = |vk-vl| vk-vl vk-vl
(vk,vl) = max(|vk-vl|,d)
Similar idea for stereo reconstruction