## 2. Probability of multiply imaged sources
In this section we briefly review the statistical concepts introduced
in K96; this also serves to define our notation. Note that with regard
to cosmogical notation we follow that of Kayser et al. (1997),
repeating here only 2 equations needed for discussion in this paper:
the comoving spherical volume element at redshift Following the K96 approach, we assume that the light deflection properties of the gravitational lenses can be modelled with a particular type of circularly symmetric lens models with a monotonically declining radial mass profile. Such lens models generally create three images and have two critical radii on which the magnification diverges (e.g. Schneider et al. 1992). It is possible to estimate the probability of the event
If the outer and inner critical angular radii of the lens potential
are respectively and
, the image magnification at radial
angular position The critical radii, the image magnifications and the source position are functions of the lens model, the luminosity of the lens galaxy and the redshifts of the source and the lens galaxy. If the underbraced functions are dropped, Eq. (3) yields the optical depth - the fraction of the sky included within the caustics of all lenses between us and the sources at redshift . The inclusion of these functions accounts for magnification bias, survey selection effects (including what is defined as a lensing event) and allows the observed image separation to be taken into account. Eq. (3) parametrically depends on
and through Eq. (1) and through the
angular size
distances, Assuming the survey selection function
by applying the parametric model (or likelihood function) where the logarithm was expanded to first order. We can combine surveys of different objects by adding the logarithms of the likelihood functions for the individual surveys, and can combine surveys containing the same objects by applying their joint selection function. In Bayesian theory the model parameters
, ,
are regarded as random quantities
with known joint prior probability density function
. Applying Bayes's theorem, the
appropriate posterior probability distribution given the observational
data where the operation `' denotes multiplication followed by normalisation. Marginalising the nuisance parameters yields the (marginal) posterior probability density function for the parameters and . In the limit where all nuisance parameters take a precise value, , the joint prior probability density function factorises into and a delta distribution , and Eq. (7) simplifies to On the basis of Eq. (7) or Eq. (8), we can calculate confidence regions for two parameters or perform further marginalisations and calculate mean values, standard deviations and marginal confidence intervals for one parameter. © European Southern Observatory (ESO) 1999 Online publication: March 29, 1999 |