Intelligent Systems (AI-2) Computer Science cpsc422 , Lecture 17 Feb, 14, 2014

1. Intelligent Systems (AI-2) Computer Science cpsc422, Lecture 17 Feb, 14, 2014 Slide Sources D. Koller, Stanford CS - Probabilistic Graphical Models D. Page, Whitehead Institute, MIT Several Figures from “Probabilistic Graphical Models: Principles and Techniques” D. Koller, N. Friedman 2009 CPSC 422, Lecture 17

2. Lecture Overview • Probabilistic Graphical models • Intro • Example • Markov Networks Representation (vs. Belief Networks) • Inference in Markov Networks (Exact and Approx.) • Applications of Markov Networks CPSC 422, Lecture 17

3. Probabilistic Graphical Models From “Probabilistic Graphical Models: Principles and Techniques” D. Koller, N. Friedman 2009 CPSC 422, Lecture 17

4. Misconception Example • Four students (Alice, Bill, Debbie, Charles) get together in pairs, to work on a homework • But only in the following pairs: AB AD DC BC • Professor misspoke and might have generated misconception • A student might have figured it out later and told study partner CPSC 422, Lecture 17

5. Example: In/Depencencies • Are A and C independent because they never spoke? • No, because A might have figure it out and told B who then told C • But if we know the values of B and D…. • And if we know the values of A and C CPSC 422, Lecture 17

6. Which of these two Bnets captures the two independencies of our example? • A. • B. • D. None • C. Both CPSC 422, Lecture 17

7. Parameterization of Markov Networks • Factors define the local interactions (like CPTs in Bnets) • What about the global model? What do you do with Bnets? X X CPSC 422, Lecture 17

8. How do we combine local models? • As in BNets by multiplying them! CPSC 422, Lecture 17

9. Multiplying Factors (same seen in 322 for VarElim) CPSC 422, Lecture 17

10. Factors do not represent marginal probs. ! • Marginal P(A,B) • Computed from the joint CPSC 422, Lecture 17

11. Step Back…. From structure to factors/potentials • In a Bnet the joint is factorized…. • In a Markov Network you have one factor for each maximal clique CPSC 422, Lecture 17

12. Directed vs. Undirected • Independencies • Factorization CPSC 422, Lecture 17

13. General definitions • Two nodes in a Markov network are independent if and only if every path between them is cut off by evidence • eg for A C • So the markov blanket of a node is…? • eg for C CPSC 422, Lecture 17

14. Lecture Overview • Probabilistic Graphical models • Intro • Example • Markov Networks Representation (vs. Belief Networks) • Inference in Markov Networks (Exact and Approx.) • Applications of Markov Networks CPSC 422, Lecture 17

15. Variable eliminationalgorithm for Bnets • To compute P(Z| Y1=v1,… ,Yj=vj) : • Construct a factor for each conditional probability. • Set the observed variables to their observed values. • Given an elimination ordering, simplify/decompose sum of products • Perform products and sum out Zi • Multiply the remaining factors Z • Normalize: divide the resulting factor f(Z) by Z f(Z) . • Variable eliminationalgorithm for Markov Networks….. CPSC 422, Lecture 17

16. Gibbs sampling for Markov Networks • Example: P(D | ¬c) • Resample non-evidence variables in a pre-defined order or a random order • Suppose we begin with A Note: never change evidence ¬c (i.e., c=0)! • What do we need to sample? • A. P(A | B) • B. P(A | B C) • C. P(B C | A) CPSC 422, Lecture 17

17. Example: Gibbs sampling • Resample probability distribution of P(A|BC) Normalized result = CPSC 422, Lecture 17

18. Example: Gibbs sampling • Resample probability distribution of B given A D Normalized result = CPSC 422, Lecture 17

19. Lecture Overview • Probabilistic Graphical models • Intro • Example • Markov Networks Representation (vs. Belief Networks) • Inference in Markov Networks (Exact and Approx.) • Applications of Markov Networks CPSC 422, Lecture 17

20. Markov Networks Applications (1): Computer Vision • Called Markov Random Fields • Stereo Reconstruction • Image Segmentation • Object recognition • Typically pairwise MRF • Each vars correspond to a pixel (or superpixel) • Edges (factors) correspond to interactions between adjacent pixels in the image • E.g., in segmentation: from generically penalize discontinuities, to road under car CPSC 422, Lecture 17

21. Image segmentation CPSC 422, Lecture 17

22. Markov Networks Applications (2): Sequence Labeling in NLP and BioInformatics • Conditional random fields (next class Wed in two weeks) CPSC 422, Lecture 17

23. CPSC 422, Lecture 17 Learning Goals for today’s class • You can: • Justify the need for undirected graphical model (Markov Networks) • Interpret local models (factors/potentials) and combine them to express the joint • Define independencies and Markov blanket for Markov Networks • Perform Exact and Approx. Inference in Markov Networks • Describe a few applications of Markov Networks

24. Next class Midterm, Mon, Feb 24, we will start at 10am sharp • How to prepare…. • Keep Working on assignment-2 ! • Office Hours next week (Wed, Fri 9-10am)x • Learning Goals (look at the end of the slides for each lecture) • Revise all the clicker questions and practice exercises • Will post more practice material next week CPSC 422, Lecture 17

25. How to acquire factors? CPSC 422, Lecture 17

26. CPSC 422, Lecture 17