= Joint coverage in detection-calibrated prevalence bounds
\b[Joint coverage in detection-calibrated prevalence bounds] controls the chance that both the observed-event bound and its detection calibration are valid. A prevalence calculation that substitutes an estimated detection probability into a sampling bound does not automatically retain the original confidence level.
In an original model with no false detections, prevalence $p$ and detection probability $q>0$ give observed-event probability $r=qp$. Let $r_U$ be an upper <confidence bound> with coverage at least $1-\alpha$, and $q_L>0$ a lower <confidence bound> with coverage at least $1-\beta$. On the event that both bounds cover,
$$
p=\frac rq\le\frac{r_U}{q_L}.
$$
The <union bound> gives $\Pr(r>r_U\text{ or }q<q_L)\le\alpha+\beta$. Hence $p_U=\min\{1,r_U/q_L\}$ has coverage at least $1-\alpha-\beta$. This argument does not require independence between the two estimates, but it does require valid marginal bounds for the stated model. If no positive lower bound on $q$ is established, the resulting prevalence bound may be uninformative.
Exact extreme-count examples follow by inverting <binomial distribution> probabilities. NIST describes exact interval construction from binomial tails and distinguishes one-sided limits.<NIST and SEMATECH e-Handbook of Statistical Methods>[§7.2.4.1, One-sided confidence intervals and Exact Intervals for Small Numbers of Failures]{ref} Zero detections among $n$ specimens gives $r_U=1-\alpha^{1/n}$. Detecting all $m$ independently known positive calibration specimens gives $q_L=\beta^{1/m}$. These are one-sided extreme-case formulas derived from $(1-r)^n$ and $q^m$.
For stipulated $n=37$, $m=20$ and $\alpha=\beta=0.025$, the combined coverage is at least 95% and $p_U\approx0.114$. Choosing two separate 95% bounds instead guarantees only at least 90% joint coverage by this argument; if their failures are independent, the joint success probability is $0.95^2=0.9025$. Neither is a 95% joint guarantee.
<Error-budget allocation between prevalence and detection calibration> need not split the allowed failure probability equally. A prespecified allocation can reflect which experiment supplies the weakest information. Calibration specimens must also represent the detection difficulty of the target specimens; a formally valid interval for an easier calibration set does not establish the relevant $q$.
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