2024 Chernoff's inequality

Chernoff's inequality

Author: tulo

August undefined, 2024

WebHoeﬀding’s inequality is a powerful technique—perhaps the most important inequality in learning theory—for bounding the probability that sums of bounded random variables are too large or too small. We will state the inequality, and then we will prove a weakened version of it based on our moment generating function calculations earlier. WebApr 18, 2024 · For an exercise, I have to prove Chernoff's inequality: P(X ≥ a) ≤ e − taMX(t) The exercise specifies that X is a random variable, MX(t) is its moment-generating function (which is finite around a small interval containing 0 ), and that I must prove that the inequality holds for every t ≥ 0.

Hoeffding

WebProof of the Chernoff bound First write the inequality as an inequality in exponents, multiplied by t>0: Pr[X<(1−δ)µ] = Pr[exp(−tX) > exp(−t(1−δ)µ)] Its not clear yet why we … WebBefore we venture into Cherno bound, let us recall Chebyshev’s inequality which gives a simple bound on the probability that a random variable deviates from its expected value … riff raff turbo boots

CS265/CME309: Randomized Algorithms and Probabilistic …

WebThis implies a multiplicative form of the Chernoff bound since: P(X n (1 + ) ) exp[ n 2 2(1 + )] and P(X n (1 ) ) exp[ n 2=2] Similar results for Bernstein and Bennet inequalities are … WebIt is constant and does not change as n increases. The bound given by Chebyshev's inequality is "stronger" than the one given by Markov's inequality. In particular, note that 4 n goes to zero as n goes to infinity. The strongest bound is the Chernoff bound. It goes to zero exponentially fast. ← previous next → WebMay 28, 2024 · the numerator on the RHS is the moment generating function of the variable. And the last inequality is the Chernoff bound. So I guess we can say that as long as there is an MGF for $X$, there is a bound. That sounds remotely useful. riff raff villains wiki

Karen Chernoff, M.D. Patient Care - Weill Cornell

Proof of Chernoff

WebConcentration Inequalities Chernoff Bounds Balls into Bins Proof of Chernoff Bounds Randomised QuickSort Lecture 5: Concentration Inequalities 24. Applications: QuickSort Quick sort is a sorting algorithm that works as following. Input:Array of different number A. Output:array A sorted in increasing order In probability theory, a Chernoff bound is an exponentially decreasing upper bound on the tail of a random variable based on its moment generating function or exponential moments. The minimum of all such exponential bounds forms the Chernoff or Chernoff-Cramér bound, which may decay faster than … See more The generic Chernoff bound for a random variable $${\displaystyle X}$$ is attained by applying Markov's inequality to $${\displaystyle e^{tX}}$$ (which is why it sometimes called the exponential Markov or exponential … See more The bounds in the following sections for Bernoulli random variables are derived by using that, for a Bernoulli random variable $${\displaystyle X_{i}}$$ with probability p of being equal to 1, See more Rudolf Ahlswede and Andreas Winter introduced a Chernoff bound for matrix-valued random variables. The following version of the inequality can be found in the work of Tropp. See more When X is the sum of n independent random variables X1, ..., Xn, the moment generating function of X is the product of the individual moment generating functions, giving that: See more Chernoff bounds may also be applied to general sums of independent, bounded random variables, regardless of their distribution; this is … See more Chernoff bounds have very useful applications in set balancing and packet routing in sparse networks. The set balancing problem arises while designing statistical … See more The following variant of Chernoff's bound can be used to bound the probability that a majority in a population will become a minority in a … See more riff raff toys auWebOct 24, 2024 · Hoeffding's inequality is a special case of the Azuma–Hoeffding inequality and McDiarmid's inequality. It is similar to the Chernoff bound, but tends to be less sharp, in particular when the variance of the random variables is small. [2] It is similar to, but incomparable with, one of Bernstein's inequalities. riff raff viper glasses

"WebChernoff bounds. Theorem 1. Suppose 0 < d, then p(X (1 +d)m) e d2m 2+d, and p(X (1 d)m) e d2m 2. You can combine both inequalities into one if you write it like this: … " - Chernoff's inequality

Hoeffding

CS265/CME309: Randomized Algorithms and Probabilistic …

Chernoff's inequality

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