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