Trigger efficiency and selection bias

Let x be an offline quantity and let epsilon of x be the probability that an eligible event is recorded under the chosen trigger policy. Multiplying the original density by that probability and renormalizing produces the recorded density. Even if a trigger uses an online estimate rather than x itself, their correlation can distort x. Finite online resolution usually turns a nominal hard threshold into an efficiency turn-on when plotted against the offline measurement.

An efficiency plateau is defined relative to a specified offline population and event environment. It can depend on direction, pileup, particle identification, and the presence of other objects. A triggered denominator is not automatically representative: requiring the tested object to fire before estimating its efficiency removes the very failures one wants to count. An orthogonal trigger or a suitable tag-and-probe construction can provide a reference, but its conditional dependencies must still be studied.

A prescale retains only a specified fraction of events satisfying a trigger condition. Live time describes availability of the acquisition system. If intrinsic efficiency, live fraction, and prescale retention are independent in the relevant conditions, their product is the total inclusion probability. Deterministic prescales can behave differently for correlated event sequences, and trigger menus can combine overlapping paths. Adding their individual efficiencies double-counts the overlap; a logical OR requires the joint acceptance information.

One can instead express exposure through an effective luminosity that already incorporates a particular live-time or prescale convention. The same loss must not then be corrected again as an efficiency factor. Conditions change during data taking, so event weights or luminosity segments should track the policy actually applied. A single average efficiency is safe only if its weighting represents the target population. Otherwise a correct global event count can coexist with a biased differential spectrum.

Inverse-probability weights become large where acceptance is small, inflating variance and sensitivity to calibration errors. Where inclusion probability is zero, recorded events provide no direct information about that population; division by zero is not a correction method. Analysts often restrict the measurement to a region with well-calibrated efficiency rather than extrapolate a threshold tail using an unconstrained model. Selection and fiducial definitions should make that choice explicit.

Selection bias also arises when thresholds, categories, or reference samples are optimized after examining an apparent signal in the same data. The eventual significance calculation must account for that procedure, not merely for the final frozen cut. Separate development data, blinded signal regions, or a predeclared optimization rule can make the sampling procedure assessable. An efficient trigger and an unbiased inference are related but distinct achievements: one concerns which events enter, while the other concerns how the complete selection rule is interpreted.