Detection probability ($P_D$)

The detection probability $P_D$ is the conditional probability that a specified procedure declares a detection when the target or signal condition is present. With $D$ the detection event and $H_1$ the signal-present condition,It depends on the signal, noise model, measurement conditions and decision rule. In a statistical hypothesis test, it is the statistical power at the specified alternative. Its complement is the probability of missing that signal; the false positive rate instead conditions on signal absence.^[ref]

High $P_D$ does not by itself imply that most detections are real. Suppose, in an invented collection of10,000 independent trial conditions,100 contain a target. A procedure detecting90% of targets and falsely alarming in5% of the9,900 target-absent conditions has expected counts90 real detections and495 false alarms. Only $90/(90+495)\approx 15.4\%$ of its alarms then correspond to targets. These are expected counts under the example model, not measured detector performance.

More generally, if $\pi =P(H_1)$ and $P_{FA}=P(D\mid H_0)$ with $H_0$ meaning absence, Bayes' theorem givesThe order of conditioning is therefore essential. Changing a decision threshold commonly trades detection against false alarms; a receiver operating characteristic displays that tradeoff.