= Near-zero probability estimates in reciprocal plug-in rules
<Near-zero probability estimates in reciprocal plug-in rules> can make a nominal randomization undefined even when the underlying probability estimate is unbiased. For an independent <measurement calibration> count $J\sim\operatorname{Binomial}(M,p)$, the empirical atom fraction $\widehat p=J/M$ has $E\widehat p=p$, but
$$
\Pr(\widehat p=0)=(1-p)^M>0\qquad(0<p<1).
$$
Thus its reciprocal is not a finite statistic on every <measurement calibration> outcome. The binomial probability mass function supplies this zero-count probability directly.<SciPy 1.16.2 documentation>[scipy.stats.binom, Notes: pmf; Methods: pmf]{ref}
In <reciprocal-estimation bias in randomized atom inclusion>, a fixed upper endpoint contributes actual probability $b$ and the auxiliary rule uses $r=1-a/\widehat p$. Valid probabilities require a separate rule when $\widehat p<a$, including zero. One possible numerical guard is $\widehat p_\epsilon=\max(J/M,\epsilon)$ with $a\le\epsilon\le1$. It makes $r=1-a/\widehat p_\epsilon$ defined and between zero and one. It does not preserve the unbiasedness or the coverage properties of the unguarded formula.
For an original finite <measurement calibration> example, take $M=20$, $p=0.02$, $a=0.005$, $b=0.995$, and $\epsilon=0.01$. A zero count occurs with probability about $0.6676$. On that outcome the guarded rule includes zero with probability $0.5$, and its included duration-model probability is $b-ap/\epsilon=0.985$, below the nominal $b-a=0.99$. Averaging over every <measurement calibration> outcome gives
$$
\sum_{j=0}^{20}\binom{20}{j}p^j(1-p)^{20-j}
\left(b-\frac{ap}{\max(j/20,\epsilon)}\right)
\approx0.9877215.
$$
The calculation assumes the future duration, <measurement calibration> sample, and auxiliary random draw have the stipulated independence, and the upper-endpoint probability is exactly $b$.
Such <finite-calibration guards for reciprocal endpoint randomization> solve a domain problem, not a coverage proof. A validated upper probability bound can provide a different conservative construction under the parent's assumptions, but an arbitrary floor is not automatically such a bound. Calibration size, lower-tail events and the actual guarded rule must all enter the final joint-probability calculation.
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