# What is the difference between Hoeffding's Inequality and Confidence Interval?

While studying Hoeffding's Inequality, I kind of tend to think it is much similar to confidence interval. I am sure and want to hear some explanations.

## 1 Answer

Confidence intervals apply to the bulk of the normal distribution: observations within a few standard deviations of the mean. So for the CLT, confidence intervals quantify the absurdity of an empirical mean by stating the intervals where 95% of the time you'd expect the empirical mean to fall in, given the null hypothesis.

However, the CLT starts to fail far away from the mean, or when the number of samples is small. Specifically, it fails to properly estimate the probability of rare events. Even though the CLT will predict very tiny probabilities for these events, the ratio of the predicted to true probability can be off by many orders of magnitude. This becomes problematic for things like risk analysis, where the CLT might predict a hurricane with chance $10^{-6}$ per day, whereas the true probability might be $10^{-3}$.

Hoeffding's inequality is one of many attempts at giving bounds on these tail events, for any distribution with bounded support, and any $n$.

• Thank you for the explanation and it helped me understand further. – user122358 Apr 3 '17 at 21:59