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Chernoff bound dependent variable

WebThe Chernoff bound is like a genericized trademark: it refers not to a particular inequality, but rather a technique for obtaining exponentially decreasing bounds on tail probabilities. … 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 … 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, One can encounter … See more Rudolf Ahlswede and Andreas Winter introduced a Chernoff bound for matrix-valued random variables. The following version of the … 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 sample, or vice versa. Suppose there is a general population A and a sub-population B ⊆ A. Mark the relative size of the … 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 known as Hoeffding's inequality. The proof follows a similar approach to the other Chernoff bounds, but applying 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

Cherno bounds, and some applications 1 Preliminaries

WebNov 23, 2024 · Siegel, A.: Towards a usable theory of Chernoff–Hoeffding bounds for heterogeneous and partially dependent random variables (manuscript) Van de Geer, … Webn be independent random variables with values in the interval r0;1s. If X X 1 X 2 X n and ErXs , then for every a ¡0 we get the bounds 1 PrrX ¥ as⁄e a 2{2n, 2 PrrX ⁄ as⁄e a 2{2n. … line the design firm uphar chibber https://ramsyscom.com

Chernoff bound - Wikipedia

WebSince the application of the Chernoff-Hoeffding bound above does not change if the subset defined by R q does not change, to prove Theorem 2.8.1 we need to show (2.3) holds … WebApr 22, 2024 · Then holding the lower and upper bounds of the numerator constant, try to get concentration bounds for the (lower(or upper) bound/denominator random variable) … WebJun 7, 2016 · How to apply Chernoff's bound when variables are not independent. Let X = ∑ i = 1 n X i, for Bernoulli random variables X i which are not necessarily independent. … line the coffers

How to obtain tail bounds for a linear combination of dependent …

Category:Lecture 2 - University of British Columbia

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Chernoff bound dependent variable

Chernoff-Hoeffding Inequality - University of Utah

WebThe critical condition that’s needed for a Chernoff bound is that the random variable be a sum of independent indicator random variables. Since that’s true for balls in bins, … WebWe seek to derive a probabilistic tool known as the Cherno Bound, a useful bound on deviation from the expected value of the sum of independent random variables. First, we …

Chernoff bound dependent variable

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WebLecture 23: Chernoff Bound & Union Bound 1 Slide Credit: Based on Stefano Tessaro’sslides for 312 19au ... Putting a limit on the probability that a random variable … 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 …

WebChernoff bound uses logarithmic number of moments, this is possible because you have n i.i.d. random variables, so their sum is very concentrated around their mean. Please note that this just my intuition (it might be very wrong ), sorry I can't provide any hard evidence. Hope that helps :-) Share Cite Follow answered Jun 22, 2012 at 9:08 dtldarek WebThus, special cases of the Bernstein inequalities are also known as the Chernoff bound, Hoeffding's inequality and Azuma's inequality . Some of the inequalities [ edit] 1. Let be independent zero-mean random variables. Suppose that almost surely, for all Then, for all positive , 2. Let be independent zero-mean random variables.

WebHoeffding, Chernoff, Bennet, and Bernstein Bounds Instructor: Sham Kakade 1 Hoeffding’s Bound We say Xis a sub-Gaussian random variable if it has quadratically bounded logarithmic moment generating func-tion,e.g. lnEe (X ) 2 2 b: For a sub-Gaussian random variable, we have P(X n + ) e n 2=2b: Similarly, P(X n ) e n 2=2b: 2 Chernoff Bound WebSection 2 we prove that the moment bound is not greater than Chernoff's bound for all distributions provided that t > 0. In Section 3 we compute the moment bound for a number of distributions, both discrete and continuous, and show that the moment bound can be substantially tighter than Chernoff's bound. In many cases of interest the order

WebThe Chernoff bound gives a much tighter control on the proba-bility that a sum of independent random variables deviates from its expectation. Although here we …

WebChernoff Bounds: Since Chernoff bounds are valid for all values of s > 0 and s < 0, we can choose s in a way to obtain the best bound, that is we can write P ( X ≥ a) ≤ min s > 0 e − s a M X ( s), P ( X ≤ a) ≤ min s < 0 e − s a M X ( s). Let us look at an example to see how we can use Chernoff bounds. Example Let X ∼ B i n o m i a l ( n, p). line the drawers refrigerator with paperWebThe Chernoff bound is like a genericized trademark: it refers not to a particular inequality, but rather a technique for obtaining exponentially decreasing bounds on tail probabilities. Much of this material comes from my CS 365 textbook, … line the cup starbucksWebOct 20, 2024 · There are several common notions of tightness of bounds, below is perhaps the simplest one. Denote the Chernoff bound as $B (x) \equiv \frac { \lambda } { \lambda - r} e^ {- rx}$ for the exponential function, which tail probability (complement CDF) is $P (X > x) = 1 - F_X (x) = e^ {-\lambda x}$. hottub 150 cmWebApr 27, 2024 · I believe one can use Hoeffding to bound $\text{Pr} \left \lbrace S_d \geq (1 + \delta) \mu_d ... Concentration bound for sum of dependent geometric random variable? 4. ... Computational indistinguishability for any distribution using a Chernoff bound. Hot Network Questions Zahlen auf Deutsch! (Numbers in German) line the designWebDec 3, 2015 · Using this representation of X it is straightforward to apply the Central limit theorem to approximate the probability P ( X ≥ 26). As I understand it, in order to apply a Chernoff bound on the probability P ( X ≥ 26) the random variable X needs to be expressed as a sum of binary random variables; the random variables X k are not binary. hot tub 1 chlorine tabletsWebrandom variables with ... R𝜖 Q2exp(−𝑘𝜖2/4) Two Extensions: 1.Dependent Random Variables 2.Sums of random matrices. Expander Chernoff Bound [AKS’87, G’94] … hot tub 220 or 110WebLecture 23: Chernoff Bound & Union Bound 1 Slide Credit: Based on Stefano Tessaro’sslides for 312 19au ... Putting a limit on the probability that a random variable is in the “tails” of the distribution (e.g., not near the middle). Usually statements in the form of … line the cup