1. Belief Propagation Kai Ju Liu March 9, 2006

2. Statistical Problems Medicine Finance Internet Computer vision

3. Inference Problems Given data B, infer A: p(A|B) Computer vision –Given image, find objects –Given two images, resolve 3D object –Given multiple images, track object

4. Conditional Probability Given event B, what is probability of A? Independence: p(A|B)=p(A) AB

5. Bayes’ Rule

6. e.g. ColdWeekday Party Hangover Marginal Probability: 8-sum Joint Probability:

7. e.g. (cont.) ColdWeekday Party Hangover Marginal Probability: 8-sum Localize probabilities:

8. Approach Define variables and connections Calculate marginal probabilities efficiently Find most likely configuration

9. Pairwise Markov Random Field 123 4 5 Basic structure: vertices, edges

10. Pairwise Markov Random Field Basic structure: vertices, edges Vertex i has set of possible states X i and observed value y i Compatibility between states and observed values, Compatibility between neighboring vertices i and j,

11. Pairwise MRF: Probabilities Joint probability: Marginal probability: –Advantage: allows average over ambiguous states –Disadvantage: complexity exponential in number of vertices

12. Belief Propagation 123 4 5

13. Beliefs replace probabilities: Messages propagate information:

14. Belief Propagation Example 13 4 5

15. BP: Questions When can we calculate beliefs exactly? When do beliefs equal probabilities? When is belief propagation efficient? Answer: Singly-Connected Graphs (SCG’s) Graphs without loops Messages terminate at leaf vertices Beliefs equal probabilities Complexity in previous example reduced from 13S 5 to 24S 2

16. BP on Loopy Graphs Messages do not terminate Possible approximate solutions –Standard belief propagation –Generalized belief propagation BP-TwoGraphs: Goals Utilize advantages of SCG’s Be accurate and efficient on loopy graphs

17. BP-TwoGraphs: SCG’s Calculate beliefs on each set of SCG’s: – Select maximum beliefs from both sets – Consider loopy graph with n vertices Select two sets of SCG’s that approximate the graph –

18. BP-TwoGraphs: Vision SCG’s Rectangular grid of pixel vertices H i : horizontal graphs G i : vertical graphs

19. Image Segmentation add noise segment

20. Image Segmentation: Results

21. Real Image Segmentation

22. Real Image Segmentation: Training

23. Real Image Segmentation: Results

24. Stereo Vision