  When searching for gravitational waves in the data from ground-based
gravitational wave detectors it is common to use a detection threshold to
reduce the number of background events which are unlikely to be the signals of
interest. However, imposing such a threshold will also discard some real
signals with low amplitude, which can potentially bias any inferences drawn
from the population of detected signals. We show how this selection bias is
naturally avoided by using the full information from the search, considering
both the selected data and our ignorance of the data that are thrown away, and
considering all relevant signal and noise models. This approach produces
unbiased estimates of parameters even in the presence of false alarms and
incomplete data. This can be seen as an extension of previous methods into the
high false rate regime where we are able to show that the quality of parameter
inference can be optimised by lowering thresholds and increasing the false
alarm rate.
