= Zero-event binomial upper confidence bound
{title2=$p_U=1-\alpha^{1/n}$}
If $X\sim\operatorname{Bin}(n,p)$ and $X=0$ is observed, the <maximum-likelihood estimate> of $p$ is zero. A one-sided $1-\alpha$ upper <confidence bound> solves $P_{p_U}(X=0)=(1-p_U)^n=\alpha$, so $p_U=1-\alpha^{1/n}$. For large $n$, $p_U\simeq-\log(\alpha)/n$, giving approximately $3/n$ at 95% confidence. This describes uncertainty after no observed events, rather than proving that the underlying <probability> is zero. It assumes independent observations with a common event probability; <clustered data> require a different uncertainty calculation.
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