Comments on the amplification bias in lens statistics

Amplification bias selection effects play an important role in lens statistics. We carefully examine the effect of amplification bias on lens statistics taking the following points into account. First, because each lens model has its own amplification bias, it must be estimated self-consistently. Secondly, in general, because the bias depends not only on quasar luminosity but also on the redshifts of the lens as well as those of the source in general, the computation of the bias and of the total optical depth cannot be separated. Finally we do not use the averaged bias over the quasar sample, but estimate the bias for each quasar in the sample. We examine the case of an isothermal sphere with and without both a finite core and a uniform sheet as our examples of lens models using the Hubble Space Telescope Snapshot Survey as the quasar sample. It is found that the conventional estimate of the bias has significant errors not only for the lensing frequency but also for the distribution of image separation angles.