A likelihood function for the Gaia Data

David W Hogg

Published 2018-04-20Version 1

When we perform probabilistic inferences with the Gaia Mission data, we technically require a likelihood function, or a probability of the (raw-ish) data as a function of stellar (astrometric and photometric) properties. Unfortunately, we aren't (at present) given access to the Gaia data directly; we are only given a Catalog of derived astrometric properties for the stars. How do we perform probabilistic inferences in this context? The answer - implicit in many publications - is that we should look at the Gaia Catalog as containing the parameters of a likelihood function, or a probability of the Gaia data, conditioned on stellar properties, evaluated at the location of the data. Concretely, my recommendation is to assume (for, say, the parallax) that the Catalog-reported value and uncertainty are the mean and root-variance of a Gaussian function that can stand in for the true likelihood function. This is the implicit assumption in most Gaia literature to date; my only goal here is to make the assumption explicit. Certain technical choices by the Mission team slightly invalidate this assumption for DR1 (TGAS), but not seriously. Generalizing beyond Gaia, it is important to downstream users of any Catalog products that they deliver likelihood information about the fundamental data; this is a challenge for the probabilistic catalogs of the future.