Bovy explained something to me about the roulette problem I was stressing about earlier in the week: If we don't have any prior information about the system, we certainly can't do inference. For example, we can't use the concept of orbital roulette if we don't think the system is bound. So we have to set up our inference such that the prior distribution function can include the information that the planets (or stars or whatever) are bound. Furthermore, we prefer dynamical solutions—in the context of roulette or mixed-phase—that have shorter dynamical times, because they will be more well mixed in angle.
This breakthrough (for me, at least) makes it clear that we can't have a Bayesian form of the problem that doesn't involve relatively complex priors, complex because the whole concept of roulette or mixed phase involves strong prior information: The information that the system is bound and long-lived (relative to its internal dynamical timescales).