Likelihood of the sky. Bayesian treatment of the geometric two-catalog matching problem

Catalogs with celestial positions and uncertainties are among the primary data products of observational astronomy. Consequently, (cross-)matching problem, namely, question how many (and which) pairs pertain to same source, remains a central part this scientific exploitation. We address matching problem for two catalogs from purely geometric point view by adopting concept processes nearest-neighbor distributions. Suitable expressions distributions were derived used model distribution spatial offsets in sky. Thus, we able estimate total number pairs, along individual probabilities between sources any uncertainties. demonstrate workings using mock apply it cross-matching X-ray eROSITA Final Equatorial-Depth survey optical Gaia counterparts.