Talk:Hoeffding's inequality

message: I think the line
:n\geq \frac{\log(2/\alpha)}{2\varepsilon^2}
in the "Confidence intervals" section doesn't follow from the previous line. If you want the minimum number of samples n such that you can be at least \textstyle (1-\alpha)% confident that \overline{X} is within \varepsilon of p, then I think you actually want
:\operatorname{P}(\ \overline{X} \notin [p-\varepsilon, p+\varepsilon]) \leq 2e^{-2\varepsilon^2n} \leq \alpha
which rearranges to give the above inequality for n.
Someone else comment: No. What is needed is the first (smallest) n which violate inequality. When  n  gets bigger, probability \alpha  (for making Type I error) must gets smaller, so probability of confidence interval become larger. Ambiguity is in words "solving the above for n gives us", when in fact what is needed is opposite of the solution to that inequality for the line mentioned. I will try to edit.  — Preceding unsigned comment added by 188.254.161.197 (talk) 18:08, 8 August 2023 (UTC)
thread_title: Mistake

message: As stated, the inequality is not true (this is easy to see for n=1).  There must be some other condition. coco
:I noticed that you added the condition t>0, which is indeed required. Is there another condition we're missing? --MarkSweep 21:17, 21 August 2005 (UTC)
replies: text: I suggest that X_i \in [0, 1] 194.25.15.170 (talk) 13:45, 16 March 2015 (UTC)

text: 79.245.218.172, could you elaborate? I've just had a look and I cannot see why the inequality would be "obviously" wrong. In particular, with your example the right-hand side is 2 \exp(-\tfrac{1}{2n}), which is greater than 1. Malparti (talk) 11:22, 17 May 2024 (UTC)
thread_title: [no title]

message: I just reverted an edit which (re)introduced a mistake in the presentation of the inequality. As stated here, the inequality involves the probability
:\Pr(S - \mathrm{E}[S] \geq nt). \qquad\textbf{correct}
Note that S is the sum of n independent random variables. This probability could also be written as
:\Pr(S/n - \mathrm{E}[S/n] \geq t), \qquad\textbf{correct}
which is how it appears in Hoeffding's paper (Theorem 2, p. 16, using slightly different notation).  In other words, Hoeffding's formulation is in terms of the mean of n independent RVs, whereas the formulation used here is in terms of their sum.  A recent edit changed this to
:\Pr(S - \mathrm{E}[S] \geq t), \qquad\textbf{wrong}
which is incorrect. --MarkSweep (call me collect) 18:50, 24 November 2005 (UTC)
thread_title: Hoeffding's theorem 2

message: Can someone point out which Bernstein inequality Hoeffding's inequality is a special case of?--Steve Kroon 14:10, 22 May 2007 (UTC)
thread_title:  Special case of Bernstein's inequality 

message: Existing version is unreadable, Article ignores dependence of results on Bennett's Inequality  —Preceding unsigned comment added by 122.106.62.116 (talk) 06:13, 27 February 2009 (UTC)
thread_title:  Hoeffding's Inequality 

message: The bounds for variables X_i seem misleading
:\Pr(X_i-\mathrm{E}[X_i] \in [a_i, b_i]) = 1. \!
Older revisions of this article (pre 2009), as well as the original paper state the bounds as 
: a_i \le X_i  \le b_i. \!
or
:\Pr(X_i \in [a_i, b_i]) = 1. \!
--18.111.112.208 (talk) 18:00, 11 December 2010 (UTC)
replies: text: It does not make any difference (I mean the first and the third version above, the second one is marginally stronger). What is misleading about it?—Emil J. 13:52, 13 December 2010 (UTC)

text: I think that the third version deals with centered versions of variable? 194.25.15.170 (talk) 13:43, 16 March 2015 (UTC)
thread_title:  Variable Bounds 

message: Can someone elaborate on the relationship to the CLT? As I understand this, these two theorems are closely related. Thanks. --138.246.2.177 (talk) 10:04, 3 December 2012 (UTC)
thread_title:  [[Central limit theorem]] 

message: I'm having a hard time finding the thing called "Hoeffding's Inequality".  Should that be at the top somewhere? 24.10.157.3 (talk) 00:06, 23 December 2021 (UTC)
replies: text: I've moved things around, it should be more clear now. Fawly (talk) 05:54, 23 December 2021 (UTC)
thread_title:  What is the inequality?