Theorem: Bounded-score confidence instance

Let ${\alpha }_{\mathrm{s}}>0$ and ${\alpha }_{\mathrm{s}}+\beta \le \alpha <1$. Define

and

With ${\left[u\right]}_{[0,1]}=\max \{0,\min \{1,u\}\}$, set

These endpoints instantiate Theorem thm:certificate with coverage at least $1-{\alpha }_{\mathrm{s}}-\beta$.

Declaring “certified” only when $U\le \epsilon$ gives

Proof. Hoeffding's inequality Hoeffding gives, conditional on the calibration record,

A union bound gives simultaneous score-mean coverage. On that event,

when $G$ also holds. Clipping preserves coverage. Integrating the conditional statistical bound and adding $\mathrm{Pr}(G^c)\le \beta$ proves the result. An invalid acceptance lies in the upper-certificate failure event.