No U-Turns for Hamiltonian Monte Carlo – comments on a paper by Hoffman and Gelman

Matthew Hoffman and Andrew Gelman recently posted a paper called “The No-U-Turn Sampler: Adaptively Setting Path Lengths in Hamiltonian Monte Carlo” on arxiv.org. It has been discussed on Andrew’s blog.

It’s a good paper, which addresses two big barriers to wider use of Hamiltonian Monte Carlo — the difficulties of tuning the trajectory length and tuning the stepsize to use when simulating a trajectory. The name “No-U-Turn Sampler” (NUTS) comes from their way of addressing the problem of tuning the trajectory length — repeatedly double the length of the current trajectory, until (simplifying a bit) there is a part of the trajectory that makes a “U-Turn”, heading back towards its starting point. This doubling method is clever, and (as I discuss below) one aspect of it seems useful even apart from any attempt to adaptively set the trajectory length. They also introduce a method of adapting the stepsize during the burn-in period, so as to achieve some desired acceptance probability.

However, I don’t think these are completely satisfactory ways of setting trajectory lengths and stepsizes. As I’ll discuss below, these problems are more complicated than they might at first appear.