Long ago, Adrian Price-Whelan (Flatiron) and I and others built The Joker, which is a Monte Carlo method (but not a MCMC method) for dealing with the Kepler problem. It exploits the fact that some parameters are linear, and some are nonlinear. This week, Lawrence Peirson (Stanford) is visiting Flatiron to generalize this point. Peirson's point is that the trick we use for linear parameters can be used for any parameters that have smooth, unimodal-ish posteriors. We just have to add some linearization and some optimization. So we are working on writing that down. And coding it up.
Along the way, Peirson found another linear parameter in The Joker, so we can now make it way, way faster. That's awesome!