1. Javier Turek and Eyal Regev

2. Several causes sharing a common effect. Y Z UiUi e+e+ e-e- X Polytrees U1U1 UnUn … … Like a tree, but with multiple parents.

3. Link matrix T x|u T x|u contains the conditional probabilities: P(X=x | U 1 =u 1,…,U n =u n ) 1. The table is huge (contains 2 n entries). 2. Who can know such information anyway? You cannot expect to find the P(X | U 1,…,U n ) table. However, you may know how every U i influences X separately.

4. The OR-gate Havent done homework Caught cheating Failed an exam More likely! OR

5. Inhibitors Failed an exam Caught cheating OR AND Paid the TA Havent done homework AND Re-doing the course

6. Inhibitors U1U1 X U2U2 UnUn I1I1 I2I2 InIn OR AND Inhibitors are independent Associate probability to an inhibitor.

7. Noisy OR-Gate U1U1 X U2U2 UnUn I1I1 I2I2 InIn OR AND

8. Noisy OR-Gate U1U1 X U2U2 UnUn I1I1 I2I2 InIn OR AND TuTu

9. Message Passing – updating X Y Z U e+e+ e-e- X

10. Y Z U e+e+ W e-e- X

11. Message Passing – updating U,Y,Z Y Z U e+e+ e-e- X

12. Y Z U e+e+ W e-e- X Message Passing – updating W,U,Y,Z

13. Y Z UiUi e+e+ e-e- X Message Passing – many parents Link matrix U1U1 UnUn … …

14. Belief Update – Noisy OR-Gate

16. Update messages Where The message to the child is the same:

17. Example Windows Vista No electricity Virus2K bug Does not start Wrong date Stolen Password Lost data D1D1 D2D2 D3D3 D4D4 M1M1 M2M2 M3M3 M4M4

18. Example D1D1 D2D2 D3D3 D4D4 M1M1 M2M2 M3M3 M4M4 0.01\0.990.1\0.90.2\0.8 0.8 0.1 0.9 0.20.5 0.1 0.7 0.8 0.2

19. Conjunction query Conjunction query q: find the belief that several events happen simultaneously. Applying the chain rule: Product of m belief updates

20. Answering a query In our example: Computing P(q):

21. Example Update the belief on M 1 Update likelihoods and priors: D1D1 D2D2 D3D3 D4D4 M1M1 M2M2 M3M3 M4M4

22. Example Update the belief on M 2 Update likelihoods and priors: D1D1 D2D2 D3D3 D4D4 M1M1 M2M2 M3M3 M4M4

23. Example Update the belief on M 3 Update likelihoods and priors: D1D1 D2D2 D3D3 D4D4 M1M1 M2M2 M3M3 M4M4

24. Example Update the belief on M 4 D1D1 D2D2 D3D3 D4D4 M1M1 M2M2 M3M3 M4M4

25. Example And the final solution is…

27. Example And the final solution is…

28. Thank You!