I spent some stolen research time today working out a simple notation for completing the square in the marginalization that Adrian Price-Whelan (Princeton), Semyeong Oh (Princeton), and I are working on for the Gaia DR1 TGAS data. It isn't hard, but you sure have to keep your head screwed on when non-square matrices are flying around, and some matrices have zero or infinite eigenvalues.
Anna Y. Q. Ho (Caltech) and I discussed things she might do at the #GaiaSprint next month. One option would be to figure out how you can infer parallax from spectrum, or spectrum from parallax. The big issue for naive approaches is that the distance or absolute magnitude uncertainties are asymmetric (think Lutz-Kelker bias and all that), but parallax uncertainties are symmetric. I suggested that we could work in the inverse-square-root-luminosity space (yes, insane) for modeling purposes and see if that helps? We would also want to use the extension of The Cannon built by Christina Eilers (MPIA) this past summer, to deal with uncertainties in the labels.