1. Probabilistic Convergence and Bounds
Junier Oliva
10-701
2/19/2013

2. Useful Inequalities
Markov’s Inequality:
Chebyshev’s Inequality:

3. Convergence in Distribution
Notation:
Definition:

4. Convergence in Pth Mean, Lp norm
Notation:
Definition:
Intuition:

5. Almost Sure Convergence
Notation:
Definition:
Intuition:

6. Convergence in Probability
Notation:
Definition:
Intuition: …

7. Relation Among Convergences

8. Convergence in Quadratic Mean => Convergence in Probability

9. Convergence in Quadratic Mean => Convergence in Probability

10. Convergence in Quadratic Mean => Convergence in Probability

11. Convergence in Quadratic Mean => Convergence in Probability

12. Convergence in Quadratic Mean => Convergence in Probability

13. Convergence in Quadratic Mean => Convergence in Probability

14. Convergence in Probability => Convergence in Distribution

15. Convergence in Probability => Convergence in Distribution

16. Convergence in Probability => Convergence in Distribution

17. Convergence in Probability => Convergence in Distribution

18. Convergence in Probability => Convergence in Distribution

19. Convergence in Probability => Convergence in Distribution

20. Convergence in Probability => Convergence in Distribution

21. Convergence in Probability => Convergence in Distribution

22. Convergence in Probability => Convergence in Distribution

23. Convergence in Probability => Convergence in Distribution

24. Convergence in Probability => Convergence in Distribution

25. Convergence in Probability => Convergence in Distribution

26. Hoeffding’s Bound
Union Bound

27. Empirical Optimization Set-up

28. Bounding Worst Case With High Probability

29. Bounding Worst Case With High Probability

30. Bounding Worst Case With High Probability

31. Bounding Worst Case With High Probability

32. Bounding Worst Case With High Probability

33. Bounding Worst Case With High Probability

34. Nearly Optimal Estimator

35. Nearly Optimal Estimator

36. Nearly Optimal Estimator

37. Nearly Optimal Estimator

38. Nearly Optimal Estimator

39. All Together

40. All Together

41. Thank You

42. Convergence in Probability => Convergence in Distribution

43. Convergence in Probability => Convergence in Distribution

44. Bounding Worst Case With High Probability

45. Bounding Worst Case With High Probability

46. Nearly Optimal Estimator