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. An efficient semi-analytical formulation for its estimation is derived which is as accurate as the standard method of numerical simulations, with only a millionth 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. Example ready-to-use scripts and some produced datasets for this method are shared in an online repository.
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