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**Chernoff Bounds, and etc.**

Presented by Kwak, Nam-ju

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**Topics A General Form of Chernoff Bounds**

Brief Idea of Proof for General Form of Chernoff Bounds More Tight form of Chernoff Bounds Application of Chernoff Bounds: Amplification Lemma of Randomized Algorithm Studies Chebyshev’s Inequality Application of Chebyshev’s Inequality Other Considerations

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**A General Form of Chernoff Bounds**

Assumption Xi’s: random variables where Xi∈{0, 1} and 1≤i≤n. P(Xi=1)=pi and therefore E[Xi]=pi. X: a sum of n independent random variables, that is, μ: the mean of X

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**A General Form of Chernoff Bounds**

When δ >0

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**Brief Idea of Proof for General Form of Chernoff Bounds**

Necessary Backgrounds Marcov’s Inequality For any random variable X≥0, When f is a non-decreasing function,

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**Brief Idea of Proof for General Form of Chernoff Bounds**

Necessary Backgrounds Upper Bound of M.G.F.

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**Brief Idea of Proof for General Form of Chernoff Bounds**

Proof of One General Case (proof)

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**Brief Idea of Proof for General Form of Chernoff Bounds**

Proof of One General Case Here, put a value of t which minimize the above expression as follows:

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**Brief Idea of Proof for General Form of Chernoff Bounds**

Proof of One General Case As a result,

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**More Tight form of Chernoff Bounds**

The form just introduced has no limitation in choosing the value of δ other than that it should be positive. When we restrict the range of the value δ can have, we can have tight versions of Chernoff Bounds.

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**More Tight form of Chernoff Bounds**

When 0<δ<1 Compare these results with the upper bo und of the general case:

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**Application of Chernoff Bounds: Amplification Lemma of Randomized Algorithm Studies**

A probabilistic Turing machine is a nondeterministic Turing machine in which each nondeterministic step has two choices. (coin-flip step) Error probability: The probability that a certain probabilistic TM produces a wrong answer for each trial. Class BPP: a set of languages which can be recognized by polynomial time probabilistic Turing Machines with an error probability of 1/3.

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**Application of Chernoff Bounds: Amplification Lemma of Randomized Algorithm Studies**

However, even though the error probability is over 1/3, if it is between 0 and 1/2 (exclusively), it belongs to BPP. By the amplification lemma, we can construct an alternative probabilistic Turing machine recognizing the same language with an error probability 2-a where a is any desired value. By adjusting the value of a, the error probability would be less than or equal to 1/3.

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**How to construct the alternative TM? (For a given input x) **

Application of Chernoff Bounds: Amplification Lemma of Randomized Algorithm Studies How to construct the alternative TM? (For a given input x) Select the value of k. Simulate the original TM 2k times. If more than k simulations result in accept, accept; otherwise, reject. Now, prove how it works.

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**Xi’s: 1 if the i-th simulation produces a wrong answer; otherwise, 0. **

Application of Chernoff Bounds: Amplification Lemma of Randomized Algorithm Studies Xi’s: 1 if the i-th simulation produces a wrong answer; otherwise, 0. X: the summation of 2k Xi’s, which means the number of wrongly answered simulations among 2k ones. ε: the error probability X~B(2k, ε) μ=E[X]=2k ε

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**Application of Chernoff Bounds: Amplification Lemma of Randomized Algorithm Studies**

P(X>k): the probability that more than hal f of the 2k simulations get a wrong answe r. We will show that P(X>k) can be less tha n 2-a for any a, by choosing k appropriatel y.

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**Here we set δ as follows:**

Application of Chernoff Bounds: Amplification Lemma of Randomized Algorithm Studies Here we set δ as follows: Therefore, by the Chernoff Bounds,

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**To make the upper bound less than or eq ual to 2-a,**

Application of Chernoff Bounds: Amplification Lemma of Randomized Algorithm Studies To make the upper bound less than or eq ual to 2-a,

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**Here, we can guarantee the right term is po sitive when 0<ε<1/2.**

Application of Chernoff Bounds: Amplification Lemma of Randomized Algorithm Studies Here, we can guarantee the right term is po sitive when 0<ε<1/2.

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**Chebyshev’s Inequality**

For a random variable X of any probabilistic distribution with mean μ and standard deviation σ, To derive the inequality, utilize Marcov’s in equality.

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**Application of Chebyshev’s Inequality**

Use of the Chebyshev Inequality To Calculate 95% Upper Confidence Limits for DDT Contaminated Soil Concentrations at a Using Chebyshev’s Inequality to Determine Sample Size in Biometric Evaluation of Fingerprint Data Superfund Site.

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**Application of Chebyshev’s Inequality**

For illustration, assume we have a large body of text, for example articles from a publication. Assume we know that the articles are on average 1000 characters long with a standard deviation of 200 characters. From Chebyshev's inequality we can then deduce that at least 75% of the articles have a length between 600 and 1400 characters (k = 2).

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Other Considerations The only restriction Markov’s Inequality impose is that X should be non-negative. It even doesn’t matter whether the standard deviation is infinite or not. e.g. a random variable X with P.D.F. it has a finite mean but a infinite standard deviation.

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Other Considerations P.D.F. E[X] Var(x)

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Conclusion Chernoff’s Bounds provide relatively nice upper bounds without too much restrictions. With known mean and standard deviation, Chebyshev’s Inequality gives tight upper bounds for the probability that a certain random variable is within a fixed distance from the mean of it.

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Conclusion Any question?

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