  The matching of sources between photometric catalogues can lead to cases
where objects of differing brightness are incorrectly assumed to be detections
of the same source. The rejection of unphysical matches can be achieved through
the inclusion of information about the sources' magnitudes. The method
described here uses the additional photometric information from both catalogues
in the process of accepting or rejecting counterparts, providing approximately
a factor 10 improvement in Bayes' factor with its inclusion. When folding in
the photometric information we avoid using prior astrophysical knowledge.
Additionally, the method allows for the possibility of no counterparts to
sources as well as the possibility that sources overlap multiple potential
counterparts. We formally describe the probability of two sources being the
same astrometric object, allowing systematic effects of astrometric
perturbation (by, e.g., contaminant objects) to be accounted for. We apply the
method to two cases. First, we test IPHAS-Gaia matches to compare the resulting
matches in two catalogues of similar wavelength coverage but differing
dynamical ranges. Second, we apply the method to matches between IPHAS and
2MASS and show that the method holds when considering two catalogues with
approximately equal astrometric precision. We discuss the importance of
including the magnitude information in each case. Additionally, we discuss
extending the method to multiple catalogue matches through an iterative
matching process. The method allows for the selection of high-quality matches
by providing an overall probability for each pairing, giving the flexibility to
choose stars known to be good matches.
