  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
unmodelled inferences on the binary black hole population. These inferences
show additional structures, specifically two peaks in the joint mass
distribution around binary masses $\sim10$ M$_\odot$ and $\sim30$ M$_\odot$.
Example ready-to-use scripts and some produced datasets for this method are
shared in an online repository.
