Richard Morey writes:
On your blog a while back, you asked why more people aren't using Hybrid (Hamiltonian) Monte Carlo. I have tried it, and found that it works quite well for many applications, but not so well for others (parameters with bounded space, and parameters with whose log-posterior has exponential functions in them, specifically). When I started using it, there wasn't much out there about it, precisely because it hasn't caught on. Well, to help remedy that a bit, I've created a CRAN package to do hybrid Monte Carlo sampling (HybridMC), and I thought this may be of interest to your readers. The back end is written in C, so it is quite fast. I've had good luck with it so far.
Cool. We should take a look at this.
I recently ran across an extension of Hamiltonian Monte Carlo which uses the Fisher information to improve performance over ordinary HMC. (The technical report has been posted to arXiv.)
I'll have to take a look at that extension to HMC. One of the annoying things about HMC is tuning the run parameters; if using the Fisher information to automate tuning is possible, that would represent a major improvement over normal HMC.
There are two other interesting extensions for HMC.
One is the ability to use the entire simulated trajectory of states for acceptance (as well as expectation estimation). See: Neal, R. M. (1994) ``An improved acceptance procedure for the hybrid Monte Carlo algorithm'', Journal of Computational Physics, vol. 111, pp. 194-203
The other is hyper-dynamic importance sampling which can allow for faster transitions between energy minima. See: http://www.cs.toronto.edu/~crismin/PAPERS/hypdyn_ivc.pdf
From my experience, Population MCMC works very well for sampling from highly multimodal distributions with multiple energy minima. This can be implemented using either standard Metropolis or Hamiltonian Monte Carlo samplers. A nice example of such a multimodal distribution is given here.
The samples from the tempered distributions can then also be used in conjunction with thermodynamic integration to calculate marginal likelihoods far more accurately than, for example, the (flawed) posterior harmonic mean method.