Bonferroni correction

Remarks

Assume each p-value is valid: if $H_i$ is true, then $P(p_i \le u) \le u$ for all $u \in [0, 1]$. Let $I_0$ be the set of true null hypotheses and $m_0$ its size. By Boole's inequality, $P(\text{some } H_i \text{ with } i \in I_0 \text{ is rejected}) \le \sum_{i \in I_0} P(p_i \le \alpha/m) \le m_0 \alpha/m \le \alpha$. So the correction controls the family-wise error rate strongly at level $\alpha$, with no assumption on the joint law of the p-values. Example: with $m = 10$ and $\alpha = 0.05$, a hypothesis is rejected when its p-value is at most $0.005$. Non-example: rejecting when $p_i \le \alpha$ is not the correction, and with $m$ independent true nulls its FWER is $1 - (1 - \alpha)^m > \alpha$. Equivalently, one rejects $H_i$ when the adjusted p-value $\min(1, m p_i)$ is at most $\alpha$.