There has been a long email conversation between Bovy, Lang, Tremaine, and me about modeling mixed-phase (ergodic) dynamical systems to determine potentials from instantaneous phase-space observations. This has been endless, for all the reasons mentioned in previous posts. I brought in Marshall before the Thanksgiving weekend, and traffic continued.
I think we understand something better than we did: We have to include parameters that explicitly model the distribution function in actions (recall that this is the action-angle formalism for integrable potentials at present). We can then marginalize over the parameters of the distribution function when we determine the parameters of the integrable potential.
But there is still the original problem that Tremaine started us on. We understand it better but we don't have a solution: How do you explicitly find the potential that is most consistent with mixed phase using straight-up Bayesian (that is, proper probabilistic) inference? It seems that all you are allowed to do is determine the posterior probability, and that posterior probability is not penalized (very much) if the particles are clustered strongly in phase. There is no way to penalize any such clustering strongly without also strongly violating the
independent and identically distributed assumption or picture in which we would like to work (for now).
Partly this all goes to show that mixed-phase systems are hard to work with. Fortunately, they also don't exist.