Probabilistic Graphical Models Representation Local Structure LogLinear Models

Probabilistic Graphical Models Representation Local Structure Log-Linear Models Daphne Koller

Log-Linear Representation • Each feature fj has a scope Dj • Different features can have same scope Daphne Koller

Representing Table Factors (X 1, X 2) = a 00 a 01 a 10 a 11 Daphne Koller

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

Ising Model Daphne Koller

Metric MRFs • All Xi take values in label space V Xi Xj want Xi and Xj to take “similar” values • Distance function : V V R – Reflexivity: (v, v) = 0 for all v – Symmetry: (v 1, v 2) = (v 2, v 1) for all v 1, v 2 – Triangle inequality: (v 1, v 2) (v 1, v 3) + (v 3, v 2) for all v 1, v 2, v 3 Daphne Koller

Metric MRFs • All Xi take values in label space V Xi Xj want Xi and Xj to take “similar” values • Distance function : V V R values of Xi and Xj far in lower probability Daphne Koller

Metric MRF Examples (vk, vl) = (vk, vl) 0 vk=vl 1 otherwise vk-vl 0 1 1 1 (vk, vl) 1 0 1 1 1 1 0 vk-vl Daphne Koller

Metric MRF: Segmentation (vk, vl) = 0 vk=vl 1 otherwise 0 1 1 1 1 0 Daphne Koller

Metric MRF: Denoising (vk, vl) = max(|vk-vl|, d) (vk, vl) = |vk-vl| vk-vl Similar idea for stereo reconstruction vk-vl Daphne Koller