第一行用到了 Chernoff bound, 第二行将SmSm替换成累加, 移出指数函数外就是累乘, 如果蓝线部分为YiYi, 可以算出它的期望和区间 (右侧蓝色部分), 随机变量YiYi满足 Hoeffding's lemma 的条件, 所以可以使用这个定理得出第三行, 公式整理后得到第四行. 这里的tt只要满足大于 0 取任何值不等式都成立, 所以令...
and then judging them as a whole. Perhaps magically, these “many simple estimates” can provide a very accurate and small representation of the large data set. The key tool in showing how
Similarly, an analogous bound holds when the Xi live on any bounded intervals [ai, bi] (and then the quantities |ai ? bi| will appear in the bound—Exercise). In fact, one can obtain similar bounds even when the Xi are unbounded, provided their distributions fall o? quickly enough, as...
(Chernoff bound) 对于随机变量 X 以及任意的 λ⩾0, 成立: E(eλX⩾eλa)⩽E(eλX)eλa. Chebyshev 不等式 对于随机变量 X∈R,记 μ=E(X), 则存在以下估计 P(|X−μ|⩾a)⩽D(X)a2. 同样利用数形结合以及示性函数的性质, 有 I_A\leqslant \frac{\left( X-\mu \right) ^2}...
can provide a very accurate and small representation of the large data set. The key tool in showing how many of these simple estimates are needed for a fixed accuracy trade-off is the Chernoff-Hoeffding inequality [2, 6]. This document provides a simple form of this bound, and two example...
can provide a very accurate and small representation of the large data set. The key tool in showing how many of these simple estimates are needed for a fixed accuracy trade-off is the Chernoff-Hoeffding inequality [2, 6]. This document provides a simple form of this bound, and two example...
Chernoff Bound与Hoeffding's Ineq 本文主要记录随机变量的Chernoff Bound和其推广Hoeffding不等式。 1.单个随机变量的Chernoff Bound 设X为实随机变量,则有: Pr(X>t)≤infs>0E(esX)estPr(X>t)≤infs>0E(esX)est 证明用Markov不等式即可。 2.多个随机变量的Chernoff Bound...