Years ago, Jo Bovy (now Toronto) and I wrote this crazy paper, in which we infer the force law in the Solar System from a snapshot of the 8 planets' positions and velocities. Because you can't infer dynamics from initial conditions in general, we had to make additional assumptions; we made the assumptions that the system is old, non-resonant, and being observed at no special time. That led to the conclusion that the distribution function should depend only on actions and not on conjugate angles.
But that's not enough: How to do inference? The frequentist solution is orbital roulette, in which you choose the force law(s) in which the conjugate angles look well mixed or uniformly distributed. That's clever, but what's the Bayesian generalization? (Or, really, specification?)
It turns out that there is no way to generate the data with a likelihood function and also insist that the angles be mixed. In Bayesian inference, all you can do is generate the data, and the data can be generated with functions that don't depend on angles. But beyond the generative model, you can't additionally insist that the angles look mixed. That isn't part of the generative model! So the solution (which was expensive) was to just model the kinematic snapshot with a very general form for the distribution function, which has a lot of flexibility but only depends on actions, generate the angles uniformly, and hope for the best. And it worked.
Why am I saying all of this? Because exactly the same issue came up today (and in the last few weeks) between Rix (MPIA) and me: I have this project to find the potential in which the chemical abundances don't vary with angle. And I can make frequentist methods that are based on minimizing angle dependences. But the only Bayesian methods I can create don't really directly insist that the abundances don't depend on angle: They only insist that the abundance distribution is controlled by the actions alone. I spent the non-discussion part of the day coding up relevant stuff.
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