WebAug 25, 2024 · 6. The Hoeffding Lemma asserts that X is a random variable bounded between [ a, b] then. E [ e λ ( X − E [ X])] ≤ e λ 2 ( b − a) 2 / 8. A typical example which asks us to show tightness of the above bound is using symmetric random variables. X s.t. X takes value a w.p. 1 / 2 and b w.p. 1 / 2. WLOG Lets take a and b to be − 1 and 1. WebExample: Hoeffding’s Inequality Proof Define A(λ) = log EeλX = log Z eλxdP(x) , where X∼ P. Then Ais the log normalization of the exponential family random variable Xλwith reference measure Pand sufficient statistic x. Since Phas bounded support, A(λ) <∞ for all λ, and we know that A′(λ) = E(Xλ), A′′(λ) = Var(Xλ).
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WebI’ll try to answer: try to write − a b − aetb + b b − aeta as a function of u = t(b − a) : this is natural as you want a bound in eu2 8. Helped by the experience, you will know that it is … WebThe proof of (20) is similar. Now we will apply Hoeffding’s inequality to improve our crude concentration bound (9) for the sum of n independent Bernoulli(µ) random variables, X1,...,Xn. Since each Xi 2 {0,1}, we can apply Theo-rem1to get, for any t ¨0, P ˆfl fl fl fl fl Xn i˘1 Xi ¡nµ fl fl fl fl fl ‚t! •2e¡2t 2/n ... cheap crewnecks for women
Understanding proof of a lemma used in Hoeffding …
In probability theory, Hoeffding's inequality provides an upper bound on the probability that the sum of bounded independent random variables deviates from its expected value by more than a certain amount. Hoeffding's inequality was proven by Wassily Hoeffding in 1963. 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 va… WebJun 25, 2024 · 1. You misinterpret the statement: the claim is that the product of S and Z − Z ′ has the same distribution as Z − Z ′. (This is true only with the additional assumptions that S has equal chances of both signs and is independent of Z, btw.) Since the values of S are in { − 1, 1 }, there are very few random variables Z for which S and ... WebProof. We have the following estimation of logarithmic moment generating function: lnEe X Ee X 1 EX+ 0:5V 2 X m=2 bm 2 m 2 = EX+ 0:5 2V(1 b) 1: The last inequality is similar to the … cutting bottom of bifold doors