Inspired by Schwab (Sternwarte Heidelberg), I spent all of Friday and a chunk of the weekend fitting models to precise radial velocity data on stars measured for the purpose of discovering exoplanets. Scwhab came to me because I had expressed confidence that an MCMC approach would be not just useful but necessary. Having said this, I had to make it work.
As always, the issue is with initialization, search, and convergence of the MCMC algorithm. The algorithm is simple and provably correct, but the proofs don't tell you how long it will take to converge to a fair sampling of the posterior distribution function. Furthermore, that convergence is a very strong function of choices you make about stepping (directions and sizes), and there is (at present) no objective way to set that. Indeed, this is a great area of research, and there are probably results there I don't know about.
One thing I did, which worked extremely well, is implement a trick suggested by Phil Marshall: I started the MCMC working on the prior alone, and slowly increased the relative weight of the likelihood, so that only after a long burn-in period was the system optimizing the true posterior. That worked extremely well. More on all this later, because all this recommends working out and writing down some
lessons learned about MCMC in practice.
[I broke the rules this week by posting only four posts. That's because on Wednesday, I got nothing done. Unfortunately, the rules demand that I admit this.]