Theorem: Pilot parameter certificate

Use $M$ independent attempted runs and let $\overline{Z}$ be the frequency of outcome zero. Define

If the actual Bernoulli mean differs from $q$ by at most $b_{\mathrm{cal}}$, then, with probability at least $1-\alpha$,

The right side may be truncated at one.

Proof. Hoeffding's inequality bounds $|\overline{Z}-q|$ by $b_{\mathrm{cal}}+\sqrt{\log (2/\alpha )/(2M)}$ with the stated coverage Hoeffding. Projection cannot increase the error from $p_{\ast }$, so $|\hat{p}-p_{\ast }|\le 2|\overline{Z}-q|$. The derivative of $p/(1-p)$ is at most nine on the specified interval.