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Probabilistic Convergence and Bounds

Probabilistic Convergence and Bounds. Junier Oliva 10-701 2/19/2013. Useful Inequalities. Markov’s Inequality: Chebyshev’s Inequality:. Convergence in Distribution. Notation: Definition:. Convergence in P th Mean, L p norm. Notation: Definition: Intuition:.

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Probabilistic Convergence and Bounds

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  1. Probabilistic Convergence and Bounds JunierOliva 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

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