Concentration inequality: Difference between revisions
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==General Chernoff Inequality== |
==General Chernoff Inequality== |
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[[Chernoff bound]] gives exponentially decreasing bounds on tail distributions of sums of independent random variables. Let <math>X_i</math> denote ''i.i.d.'' random variables, satisfying <math>X_i \geq E(X_i)-a_i-M</math>, for <math>1 \leq i \leq n</math>.<math>X = \sum_{i=1}^n X_i</math> we have |
[[Chernoff bound]] gives exponentially decreasing bounds on tail distributions of sums of independent random variables. Let <math>X_i</math> denote ''i.i.d.'' random variables, satisfying <math>X_i \geq E(X_i)-a_i-M</math>, for <math>1 \leq i \leq n</math>. |
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: <math>X = \sum_{i=1}^n X_i</math> we have lower tail inequality: |
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: <math> |
: <math> |
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\Pr[X \leq E(X)-\lambda]\leq e^{-\frac{\lambda^2}{2(Var(X)+\sum_{i=1}^n a_i^2+M\lambda/3)}} |
\Pr[X \leq E(X)-\lambda]\leq e^{-\frac{\lambda^2}{2(Var(X)+\sum_{i=1}^n a_i^2+M\lambda/3)}} |
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</math> |
</math> |
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If <math>X_i</math> satisfies <math>X_i \geq E(X_i)+a_i+M</math>, we have upper tail inequality: |
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: <math> |
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\Pr[X \leq E(X)+\lambda]\leq e^{-\frac{\lambda^2}{2(Var(X)+\sum_{i=1}^n a_i^2+M\lambda/3)}} |
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</math> |
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If <math>X_i</math> are ''i.i.d.'', <math>|X_i| \leq 1</math> and <math>\sigma^2</math> is the variance of <math>X_i</math>. A typical version of Chernoff Inequality is: |
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: <math> |
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\Pr[|X| \geq k\sigma]\leq 2e^{-k^2/4} |
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</math> |
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==Hoeffdin's Inequality== |
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[[Hoeffdin's inequality]] can be stated as follows: |
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If :<math>X_1, \dots, X_n \!</math> are independent. Assume that the <math>X_i</math> are [[almost sure]]ly bounded; that is, assume for <math>1 \leq i \leq n</math> that |
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:<math>\Pr(X_i \in [a_i, b_i]) = 1. \!</math> |
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Then, for the empirical mean of these variables |
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:<math>\overline{X} = (X_1 + \cdots + X_n)/n \!</math> |
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we have the inequalities (Hoeffding 1963, Theorem 2 <ref>Wassily Hoeffding, Probability inequalities for sums of bounded random variables, ''Journal of the American Statistical Association'' '''58''' (301): 13–30, March 1963. ([http://links.jstor.org/sici?sici=0162-1459%28196303%2958%3A301%3C13%3APIFSOB%3E2.0.CO%3B2-D JSTOR]) |
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</ref>): |
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:<math>\Pr(\overline{X} - \mathrm{E}[\overline{X}] \geq t) \leq \exp \left( - \frac{2t^2n^2}{\sum_{i=1}^n (b_i - a_i)^2} \right),\!</math> |
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:<math>\Pr(|\overline{X} - \mathrm{E}[\overline{X}]| \geq t) \leq 2\exp \left( - \frac{2t^2n^2}{\sum_{i=1}^n (b_i - a_i)^2} \right),\!</math> |
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==References== |
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{{reflist}} |
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[[Category:Mathematics]] |
[[Category:Mathematics]] |
Revision as of 03:16, 9 June 2011
In mathematics, concentration inequalities provide probability bounds on how a random variable deviates from some value (e.g. its expectation). The laws of large numbers of classical probability theory state that sums of independent random variables are, under very mild conditions, close to their expectation with a large probability. Such sums are the most basic examples of random variables concentrated around their mean. Recent results shows that such behavior is shared by other functions of independent random variables.
Markov's inequality
If X is any random variable and a > 0, then
Proof can be found here.
We can extend Markov's inequality to a strictly increasing and non-negative function . We have
Chebyshev's inequality
Chebyshev's inequality is a special case of generalized Markov's inequality when
If X is any random variable and a > 0, then
Where Var(X) is the variance of X, defined as:
Asymptotic Behavior of Binomial Distribution
If a random variable X follows the binomial distribution with parameter and . The probability of getting exact successes in trials is given by the probability mass function
Let and 's are i.i.d. Bernoulli random variables with parameter . follows the binomial distribution with parameter with parameter and . Central Limit Theorem suggests when , is approximately normally distributed with mean and variance , and
For , where is a constant, the limit distribution of binomial distribution is the Poisson distribution
General Chernoff Inequality
Chernoff bound gives exponentially decreasing bounds on tail distributions of sums of independent random variables. Let denote i.i.d. random variables, satisfying , for .
- we have lower tail inequality:
If satisfies , we have upper tail inequality:
If are i.i.d., and is the variance of . A typical version of Chernoff Inequality is:
Hoeffdin's Inequality
Hoeffdin's inequality can be stated as follows:
If : are independent. Assume that the are almost surely bounded; that is, assume for that
Then, for the empirical mean of these variables
we have the inequalities (Hoeffding 1963, Theorem 2 [1]):