scope_rl.utils.estimate_confidence_interval_by_empirical_bernstein#
- scope_rl.utils.estimate_confidence_interval_by_empirical_bernstein(samples, alpha=0.05, **kwargs)[source]#
Estimate the confidence interval by the empirical bernstein inequality.
Note
The empirical bernstein inequality provides high-probability bounds of the expectation \(\mu := \mathbb{E}[X], X \sim p(X)\) as follows.
\[|\hat{\mu} - \mu| \leq \frac{7 X_{\max} \log(2 / \alpha)}{3 (n - 1)} + \sqrt{\frac{2 \hat{\mathbb{V}}(X) \log(2 / \alpha)}{n(n - 1)}},\]which holds with probability \(1 - \alpha\) where \(n\) is the data size and \(\hat{\mathbb{V}}\) is the sample variance.