Probabilistic Convergence and Bounds
Junier Oliva10-701
2/19/2013
Useful Inequalities
• Markov’s Inequality:
• Chebyshev’s Inequality:
Convergence in Distribution
• Notation:• Definition:
Convergence in Pth Mean, Lp norm
• Notation:• Definition:• Intuition:
Almost Sure Convergence
• Notation:• Definition:• Intuition:
Convergence in Probability• Notation:• Definition:
• Intuition:
…
Relation Among Convergences
Convergence in Quadratic Mean => Convergence in Probability
Convergence in Quadratic Mean => Convergence in Probability
Convergence in Quadratic Mean => Convergence in Probability
Convergence in Quadratic Mean => Convergence in Probability
Convergence in Quadratic Mean => Convergence in Probability
Convergence in Quadratic Mean => Convergence in Probability
Convergence in Probability => Convergence in Distribution
Convergence in Probability => Convergence in Distribution
Convergence in Probability => Convergence in Distribution
Convergence in Probability => Convergence in Distribution
Convergence in Probability => Convergence in Distribution
Convergence in Probability => Convergence in Distribution
Convergence in Probability => Convergence in Distribution
Convergence in Probability => Convergence in Distribution
Convergence in Probability => Convergence in Distribution
Convergence in Probability => Convergence in Distribution
Convergence in Probability => Convergence in Distribution
Convergence in Probability => Convergence in Distribution
Hoeffding’s Bound
Union Bound
Empirical Optimization Set-up
Bounding Worst Case With High Probability
Bounding Worst Case With High Probability
Bounding Worst Case With High Probability
Bounding Worst Case With High Probability
Bounding Worst Case With High Probability
Bounding Worst Case With High Probability
Nearly Optimal Estimator
Nearly Optimal Estimator
Nearly Optimal Estimator
Nearly Optimal Estimator
Nearly Optimal Estimator
All Together
All Together
Thank You
Convergence in Probability => Convergence in Distribution
Convergence in Probability => Convergence in Distribution
Bounding Worst Case With High Probability
Bounding Worst Case With High Probability
Nearly Optimal Estimator