John Salvatier pointed me to this blog on derivative based MCMC algorithms (also sometimes called “hybrid” or “Hamiltonian” Monte Carlo) and automatic differentiation as the future of MCMC.
This all makes sense to me and is consistent both with my mathematical intuition from studying Metropolis algorithms and my experience with Matt using hybrid MCMC when fitting hierarchical spline models. In particular, I agree with Salvatier’s point about the potential for computation of analytic derivatives of the log-density function. As long as we’re mostly snapping together our models using analytically-simple pieces, the same part of the program that handles the computation of log-posterior densities should also be able to compute derivatives analytically.
I’ve been a big fan of automatic derivative-based MCMC methods since I started hearing about them a couple years ago (I’m thinking of the DREAM project and of Mark Girolami’s paper), and I too wonder why they haven’t been used more. I guess we can try implementing them in our current project in which we’re trying to fit models with deep interactions. I also suspect there are some underlying connections between derivative-based jumping rules and redundant parameterizations for hierarchical models.
It’s funny. Salvatier is saying what I’ve been saying (not very convincingly) for a couple years. But, somehow, seeing it in somebody else’s words makes it much more persuasive, and again I’m all excited about this stuff.
My only amendment to Salvatier’s blog is that I wouldn’t refer to these as “new” algorithms; they’ve been around for something like 25 years, I think.