The observed distributions of the source properties from gravitational-wave detections are biased due to the selection effects and detection criteria in the detections, analogous to the Malmquist bias. In this work, this observation bias is investigated through its fundamental statistical and physical origins. A semi-analytical formulation for its estimation for a network of detectors is derived which is more efficient and precise than the standard method of numerical simulations, with only a fraction of the computational cost. Then, the estimated bias is used for model-independent inferences on the binary black hole population. These inferences show additional structures, specifically two potential mass gaps in the joint mass distribution, which were not found via modelled inferences.
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