1. An introduction to Bayesian networks Stochastic Processes Course Hossein Amirkhani Spring 2011

2. 2 An introduction to Bayesian networks Outline  Introduction,  Bayesian Networks,  Probabilistic Graphical Models,  Conditional Independence,  I-equivalence.

3. 3 An introduction to Bayesian networks Introduction

4. 4 An introduction to Bayesian networks Bayesian Networks

5. 5 An introduction to Bayesian networks Probabilistic Graphical Models  Nodes are the random variables in our domain.  Edges correspond, intuitively, to direct influence of one node on another. Factor GraphMarkov Random FieldBayesian Network

6. 6 An introduction to Bayesian networks Probabilistic Graphical Models Graphical models = statistics × graph theory × computer science.

7. 7 An introduction to Bayesian networks Bayesian Networks

8. 8 An introduction to Bayesian networks Bayesian Networks

9. 9 An introduction to Bayesian networks Conditional Independence: Example 1 tail-to-tail at c

10. 10 An introduction to Bayesian networks Conditional Independence: Example 1

11. 11 An introduction to Bayesian networks Conditional Independence: Example 1 Smoking Lung Cancer Yellow Teeth

12. 12 An introduction to Bayesian networks Conditional Independence: Example 2 head-to-tail at c

13. 13 An introduction to Bayesian networks Conditional Independence: Example 2

14. 14 An introduction to Bayesian networks Conditional Independence: Example 2 Type of Car SpeedAmount of speeding Fine

15. 15 An introduction to Bayesian networks Conditional Independence: Example 3 head-to-head at c v-structure

16. 16 An introduction to Bayesian networks Conditional Independence: Example 3

17. 17 An introduction to Bayesian networks Conditional Independence: Example 3 Ability of team A Ability of team B Outcome of A vs. B game

18. 18 An introduction to Bayesian networks D-separation

19. 19 An introduction to Bayesian networks I-equivalence

20. 20 An introduction to Bayesian networks The skeleton of a Bayesian network

21. 21 An introduction to Bayesian networks Immorality

22. 22 An introduction to Bayesian networks Relationship between immorality, skeleton and I-equivalence

23. 23 An introduction to Bayesian networks Identifying the Undirected Skeleton

24. 24 An introduction to Bayesian networks Identifying the Undirected Skeleton

25. 25 An introduction to Bayesian networks

26. 26 An introduction to Bayesian networks Identifying Immoralities

27. 27 An introduction to Bayesian networks

28. 28 An introduction to Bayesian networks Representing Equivalence Classes

29. 29 An introduction to Bayesian networks Representing Equivalence Classes

30. 30 An introduction to Bayesian networks Representing Equivalence Classes  Is the output of Mark-Immoralities the class PDAG?  Clearly, edges involved in immoralities must be directed in K.  The obvious question is whether K can contain directed edges that are not involved in immoralities.  In other words, can there be additional edges whose direction is necessarily the same in every member of the equivalence class?

31. 31 An introduction to Bayesian networks Rules

32. 32 An introduction to Bayesian networks

33. 33 An introduction to Bayesian networks Example

34. 34 An introduction to Bayesian networks References  D. Koller and N. Friedman: Probabilistic Graphical Models. MIT Press, 2009.  C. M. Bishop: Pattern Recognition and Machine Learning. Springer, 2006.

35. 35 An introduction to Bayesian networks