I spent time today writing in the method section of the Anderson et al paper. I realized in writing it that we have been thinking about our model of the color–magnitude diagram as being a prior on the distance or parallax. But it isn't really, it is a prior on the color and magnitude, which for a given noisy, observed star, becomes a prior on the parallax. We will compute these implicit priors explicitly (it is a different prior for every star) for our paper output. We have to describe this all patiently and well!
At some point during the day, Jo Bovy (Toronto) asked a very simple question about statistics: Why does re-sampling the data (given presumed-known Gaussian noise variances in the data space) and re-fitting deliver samples of the fit parameters that span the same uncertainty distribution as the likelihood function would imply? This is only true for linear fitting, of course, but why is it true (and no, I don't mean what is the mathematical formula!)? My view is that this is (sort-of) a coincidence rather than a result, especially since it (to my mind) confuses the likelihood and the posterior. But it is an oddly deep question.