Lang and I hacked away at our Comet Holmes project today, trying to get an MCMC to find the global optimum of the likelihood function, and sample it. We know the right answer—the point is not to get the right answer in this case—the point is to make a system that just works, every time. In this and in my exoplanet stuff, there are many local minima; of course that is true for generic optimization problems, but I have an intuition that a lot of the simple dynamics problems have similar kinds of local minima.
One theme of these kinds of problems is that we (meaning astronomers) tend to use MCMC for three things that are really distinct operations:
- We want the MCMC to crawl the parameter space and find, among all the local optima, the global optimum. This is search.
- We want the MCMC to gradient-descend into that global optimum. This is optimization.
- We want the MCMC to give us back a fair sampling from the likelihood or posterior probability distribution function. This is integration.