bayesian orbital roulette

While in Princeton, Tremaine assigned Bovy and me the problem of making a Bayesian formulation of the orbital roulette problem. The existing formulation is disturbingly frequentist (choose a statistic and optimize it!), but Tremaine thought there might be problems in principle with any Bayesian formulation. Conversations between Bovy, Lang, and myself got us to a solution that works beautifully! Unfortunately, we don't quite understand the justification of our own method, which is an odd situation, but maybe not all that uncommon when inference gets hard. This is a baby problem for the Gaia problem that I have been talking about, so it is nice to already have something solid to talk about.

The idea in the Bayesian formulation is that the fact that you expect the phases of the orbits to be randomly distributed (that is, you expect a flat distribution of phases) turns into a distribution of inferred potentials (because, for each observed position and velocity, the phase is a function of what potential you choose). The problem with our formulation is that we used this thinking to jump to the solution, without going through all the steps that ought to be there. More when we have it.