1 / 20

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

herman
Download Presentation

Probabilistic graphical models

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Probabilistic graphical models

  2. 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

  3. Classification of probabilistic graphical models Both directed and undirected arcs:chain graphs

  4. 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 D E F G

  5. 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

  6. Training Bayesian network: frequencies Known: frequencies  Pr(c, s, r, w) for all (c, s, r, w)

  7. 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

  8. Application: modeling DNA motifs • Profile model: no dependences between positions • Markov model: dependence between adjacent positions • Bayesian network model: non-local dependences

  9. 1 2 3 4 5 6 A1 A2 A3 A4 A5 A6 A DNA profile TATAAA TATAAT TATAAA TATAAA TATAAA TATTAA TTAAAA TAGAAA 1 2 3 4 5 6 T 8 1 6 1 0 1 C 0 0 0 0 0 0 A 0 7 1 7 8 7 G 0 0 1 0 0 0 The nucleotide distributions at different sites are independent !

  10. m4 m2 m1 m5 11 15 14 12 Mixture of profile model A1 A3 A4 A5 A6 A2 Z The nt-distributions at different sites are conditionally independent but marginally dependent !

  11. 1 2 3 4 5 6 A1 A2 A3 A4 A5 A6 Tree model The nt-distributions at different sites are pairwisely dependent !

  12. 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

  13. 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 A C B D

  14. Markov network • Represents joint distribution • Unnormalized factor • Partition function • Probability A C B D

  15. 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)

  16. Maximal cliques • {A,B} • {B,C} • {C,D} • {A,D} Maximal cliques • {A,B,C} • {A,C,D} Markov Network Factors A A B D B D C C

  17. 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

  18. Application: Image analysis • The image segmentation problem • Task: Partition an image into distinct parts of the scene • Example: separate water, sky, background

  19. 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 potentialexp(1{Xi=Xj}) for >0

  20. Markov Network for Segmentation Appearance distribution • Solution: inference • Find most likely assignment to Xi variables X11 X12 X13 X14 X21 X22 X23 X24 X31 X32 X33 X34 Contiguity preference

More Related