Once again I got confused about our probabilistic inference translation of the frequentist orbital roulette technique for inferring the dynamical properties of systems for which you only have a kinematic snapshot. The key issue is that the frequentist gets to choose which statistic he or she wants to test; he or she only needs a statistic which will assign low likelihood to unlikely data sets given the assumptions or model. The Bayesian, on the other hand, must treat the observations as modifying prior probability distributions for parameters (or hypotheses) into posterior probability distributions. The only
freedom is in choosing the prior, but that isn't even a freedom if you are a strict Bayesian (that is, if you use your prior probability distribution functions to accurately represent your true prior knowledge).
Now, if I could convince myself of the following I would be happy: When you see a snapshot of a dynamical system about which you know nothing except that it is long-lived, your true (I mean accurate) prior probability distribution over actions, angles (think phase space coordinates for an integrable dynamical system here), and dynamical parameters (think parameters of the gravitational potential, such as the total mass and dependence on radius here) is such that whatever values of position and velocity you observed, your Bayes-theorem-generated posterior probability distribution for the angles is nonetheless perfectly flat.
I have called this choice of prior the
roulette prior since if we adopt it, we get a very nice Bayesian formulation of orbital roulette. But I can't quite convince myself that this is really an accurate representation of one's prior knowledge.