An Expectation Maximisation Algorithm for One-to-Many Record Linkage, Illustrated on the Problem of Matching Far Infra-Red Astronomical Sources to Optical Counterparts

The problem of record linkage is often seen simply in terms of making links between data points that might be generated from the same source. However, in many cases the grounds for linking items is itself not certain. In fact it is often desirable to learn, in an unsupervised manner, what form linked objects take in different databases. One simple case of this is the ``one to many'' linkage problem, where each object in one dataset is potentially linked to one of many objects in another dataset, and where the candidate matches are mutually exclusive. We show how the Expectation Maximisation algorithm can be used for this matching problem, both to calculate the probability of a match, and to learn something about the characteristics that matched objects have. The approach is derived for the specific astronomical problem of linking far infra-red observations to optical counterparts, but is generally applicable. This report outlines the theory of this record linkage procedure, but does not discuss its application or any implementational details.