# Empirical Bernstein Confidence Bound

** Published:**

A small note for a useful conclusion for giving concentration bound.

## Theorem on Empirical Bernstein Confidence Bound

The assumption of bounded entries in matrix allows us to use exponential concentration inequalities and construct tight finite-sample confidence bounds. Based on the following empirical Bernstein inequality in Maurer and Pontil, some useful results might be obtained.

**Theorem** Let Z, Z_{1}, …, Z_{n} be i.i.d. random variables with values in [0, 1] and let δ > 0. Then with probability at least 1 − δ in the i.i.d. vector **Z** = (Z_{1}, · · · , Z_{n}) that

where V_{n}(**Z**) is the sample variance.

This theorem looks much useful and I am quite curious about its proof. For analysis on Stein lower bound, we might first obtain the confidence bound for entries in a matrix whose trace of inverse is closely related to our Stein lower bound.