= Detection-threshold selection bias in pulse-energy means
\b[Detection-threshold selection bias in pulse-energy means] occurs when detected pulses have a different energy distribution from all emitted pulses. A detector conditions the reported mean on detection, not merely on emission. Let $U$ be emitted-pulse energy with finite mean and let $s(u)=\Pr(D\mid U=u,E)$ be detection probability conditional on emission $E$. Then
$$
\mathbb E[U\mid D,E]=\frac{\mathbb E[U s(U)\mid E]}{\mathbb E[s(U)\mid E]},\qquad
\mathbb E[U\mid D,E]-\mathbb E[U\mid E]
=\frac{\operatorname{Cov}(U,s(U)\mid E)}{\mathbb E[s(U)\mid E]}.
$$
The denominator must be positive. An ideal energy threshold has $s(u)=1$ above threshold and zero below. Its nondecreasing selection preferentially retains stronger pulses; false triggers or selection on another variable require a different model.
For an original distribution, emitted energies are $1$, $3$ and $7\,\mathrm{nJ}$ with probabilities $0.25$, $0.50$ and $0.25$. Their <conditional pulse-energy mean given emission> is $3.5\,\mathrm{nJ}$. A perfect receiver accepting only energies above $2\,\mathrm{nJ}$ detects $75\%$ of emitted pulses and reports
$$
\frac{0.50\cdot3+0.25\cdot7}{0.75}=\frac{13}{3}\,\mathrm{nJ}\approx4.33\,\mathrm{nJ}.
$$
The detected-event mean is about $23.8\%$ too high for the emitted-pulse mean, even with perfect energy measurements on every accepted event.
A trigger counter, a lower-threshold reference monitor and a threshold sweep can distinguish missing detections from missing emissions. Historical calorimetric inference uses total energy, exposure time and pulse rate; the rate must describe the population whose mean is requested.<Calibration Service for Laser Power and Energy at 248 nm by Rodney W. Leonhardt>[§6.2, Laser Stability, calorimetric pulse-energy inference from energy, injection time and pulse rate]{ref} This selection example is an independent probability model, not a reconstruction of that experiment.
Known positive detection probabilities can support <inverse probability weighting>. A hard threshold with zero probability below it leaves that part of the energy distribution unidentified from detected events alone. Reporting the acceptance fraction as well as the detected-event mean makes the conditioning explicit; it does not recover the unseen weak-pulse energies.
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