Identifying the infrared counterparts of X-ray sources in Galactic plane fields such as those of the MYStIX project presents particular difficulties due to the high density of infrared sources. This high stellar density makes it inevitable that a large fraction of X-ray positions will have a faint field star close to them, which standard matching techniques may incorrectly take to be the counterpart. Instead we use the infrared data to create a model of both the field star and counterpart magnitude distributions, which we then combine with a Bayesian technique to yield a probability that any star is the counterpart of an X-ray source. In our more crowded fields, between 10% and 20% of counterparts that would be identified on the grounds of being the closest star to an X-ray position within a 99% confidence error circle are instead identified by the Bayesian technique as field stars. These stars are preferentially concentrated at faint magnitudes. Equally importantly the technique also gives a probability that the true counterpart to the X-ray source falls beneath the magnitude limit of the infrared catalog. In deriving our method, we place it in the context of other procedures for matching astronomical catalogs.