chemicals across the Milky Way, combinatoric degeneracies

A few weeks ago, Jason Sanders (Cambridge) computed isochrone-based distances for all the red giants for which Andy Casey, Melissa Ness, and I measured 15-dimensional chemical abundance information. Finally today I incorporated these distances, computed Galactocentric positions (approximately) and plotted chemical abundance trends in the Galaxy. They look super-promising! Here's one:

I also had a long phone call with Dan Foreman-Mackey about various things. We had a good idea on the call for testing MCMC methods in problems where there are labeling degeneracies. For example, if you do a 5-planet model for a stellar lightcurve, you can make an identical model by swapping all of the properties of the third and fourth planet in your list. That's a combinatoric degeneracy: It means that there are (at least) 5-factorial identical modes in your posterior pdf. We figured out a large family of analytic distributions with these properties, and even ones where we can compute (analytically) the fully marginalized likelihood (FML) or Bayesian evidence integral. This will give us benchmarks for testing codes that claim to compute this notoriously hard integral.