There has been some interesting traffic on the PyMC mailing list lately. It seems that there is a common trouble with the “Adaptive Metropolis” step method, and it’s failure to converge. I’ve been quite impressed with this approach, and I haven’t had the problems that others reported, so I started wondering: Have I been lucky? Have I been not looking carefully?
I decided to do some experiments to make Metropolis and Adaptive Metropolis shine, and since I’m an aspiring math filmmaker these days, I made my experiments into movies.
I consider the Metropolis step method the essence of MCMC. You have a particular point in parameter space, and you tentatively perturb it to a new random point, and then decide to accept or reject this new point with a carefully designed probability, so that the stationary distribution of the Markov chain is something you’re interested in. It’s magic, but it’s good, simple magic.
Here is the simplest example I could come up with, sampling from a uniform distribution on the unit square in the plane using Metropolis steps. Any self-respecting step method can sample from the unit square in two dimensions!