Just a brief note to point out this recent arXival by Leistedt & Hogg in which the color-magnitude diagram is used to help refine (parallax) distance estimates from Gaia via a Bayesian hierarchical model. The idea is that although a star’s apparent magnitude and colour tell us very little about its distance in isolation they can help to provide a substantial amount of Bayesian shrinkage (via the absolute magnitude-colour diagram) given a large enough collection of stars with additional noisy distance measurements (that one may hope to refine). Bayesian shrinkage (see David van Dyk’s talk from IAUS306) is a powerful idea in hierarchical modelling and, although now used fairly widely in photometric redshift studies and galaxy classifications, its potential certainly has not been exhausted in any sense.
What I found particularly interesting here was the form chosen to represent the density of (true/latent) stellar positions on the (absolute) colour-magnitude diagram, which was a Normal/Gaussian mixture model in which the means of each component are placed in a grid over the 2D parameter space, the standard deviations of the components are all held fixed, and only the vector of component weights is to be inferred. The motivation given for this modelling decision is that it renders the inference process easier, in particular because it makes it easier to marginalise out the latent variables in the Gibbs sampling step. In fact the benefits of such a model run even deeper by way of conferring convexity to the mixture model construction as known from studies of this model from the perspective non-parametric maximum likelihood estimation (e.g. Feng & Dicker 2016).
Edit: While cleaning up my hundreds of open browser tabs I realised I should also have pointed to this recent arXival by Si et al. as another example of hierarchical Bayesian shrinkage for Gaia data.