A likelihood function for the Gaia Data

TL;DR

This paper discusses how to treat Gaia Catalog data as an implicit likelihood function for probabilistic inference, emphasizing the importance of explicit likelihood representations for future catalogs.

Contribution

It explicitly formalizes the common assumption that Gaia Catalog values represent likelihood functions, clarifying the implicit statistical interpretation used in current literature.

Findings

Gaia Catalog values can be treated as Gaussian likelihoods for inference

This assumption is generally valid despite minor technical issues in DR1

Explicit likelihood representations are crucial for future probabilistic catalogs

Abstract

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.