Avoiding selection bias : A unified treatment of thresholded data

When searching for populations of rare and/or weak signals in noisy data, it is common to use a detection threshold to remove marginal events which are unlikely to be the signals of interest; or a detector might have limited sensitivity, causing it to not detect some of the population. In both cases a selection of data has occurred, which can potentially bias any inferences drawn from the remaining data, and this effect must be corrected for. We show how the 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.