also considered the case of counterparts in one single primary catalogue.

For the association of entire counterpart sets with the X-ray sources within a search radius of 10, we use two estimators. In one case, we computed the simple probability of chance coincidence according to [@1986MNRAS.218...31D] (done purely as a cross-check and for reference since it is what was used for the `XLSS` and `2XLSS` catalogues): $$\label{EqP}
  P = 1 - exp(-\pi~ n(<m)~  d^2 )$$

In the other case, we used the more robust likelihood ratio estimator [@1992MNRAS.259..413S], which has been used for other surveys, e.g. [@2007ApJS..172..353B]: $$\label{EqLR}
  LR = (q(m) exp(-0.5 d^2/\sigma^2)/2\pi\sigma^2) / K n(m)$$

In both formulae, $d$ is the distance in arcsec between the X-ray position and the candidate counterpart position, the positional error $\sigma$ is assumed for simplicity equal to 1, while the dependency on the magnitude $m$ of the counterpart is either via the sky density (sources per square arcsec) $n(<m)$ of objects brighter than $m$, or the sky density per magnitude bin $n(m)$. The term $q(m)$ in Eq. \[EqLR\] is defined as $$\label{EqQ}
  q(m) = g(m) - K n(m)
,$$ where $g(m)$ is the sky density of putative true counterparts, i.e. unique objects within 