[In a violation of the rules (at right), I haven't posted for a couple of days. Not much research got done, between finishing a funding proposal, giving a talk at Michigan State University (thanks everyone!), and the related travel.]
Bovy, Lang, and I had the realization that, in the one-dimensional simple-harmonic-oscillator formulation of the Bayesian orbital roulette problem, the priors matter. Some apparently natural choices of prior on amplitude and frequency make it such that the marginalized posterior probability distribution function for the phase is not the same as the prior distribution, that is, flat in phase. These choices of prior must be wrong choices in the absence of truly informative prior information, because the phases ought to be uniformly distributed no matter what.
Once we figured out which priors are truly uninformative, we were able to make the whole thing work. Time to write it up and extend it to more complicated one-dimensional cases (such as Oort's method for determination of the vertical mass distribution in the disk).