MCMC sampling on Riemann manifolds

This week in lab meeting, we discussed MCMC methods presented in

Mark Girolami & Ben Calderhead
Riemann manifold Langevin and Hamiltonian Monte Carlo methods
Journal of the Royal Statistical Society: Series B

This paper gives a new way to improve the existing Metropolis adjusted Langevin algorithm (MALA) and Hamiltonian Monte Carlo (HMC) by taking advantage of the geometric structure inherent in the problem. MALA uses a known diffusion process that convergences to a target distribution to propose steps in a Metropolis-Hastings setup. The authors show that this diffusion process can be defined on a Riemann manifold and, by providing an appropriate metric for the manifold, the sampler will automatically adjust its movement through the probability space to better match the target distribution.  The Fisher information turns out to be a natural and useful metric. This method could be useful in many sampling problems since it can provide quicker mixing than simpler methods, such as random-walk Metropolis, while getting rid of the need to hand-tune carefully and painstakingly the sampler in standard HMC and MALA. This of course comes with additional computational costs and it requires more math to set up than other samplers.

The math and physics behind these methods (differential geometry, Hamiltonian mechanics, stochastic differential equations) were a little difficult for us to tackle over two 1-hour lab meetings, but this paper gives good description on how to implement Riemann Manifold HMC without forcing readers to spend a year learning differential geometry. This could be a nice tool to have ready whenever our projects require some sampling methods.