## non-reversible Langevin samplers

**I**n the train to Oxford yesterday night, I read through the recently arXived Duncan et al.’s *Nonreversible Langevin Samplers: Splitting Schemes, Analysis and Implementation*. Standing up the whole trip in the great tradition of British trains.

The paper is fairly theoretical and full of Foster-Lyapunov assumptions but aims at defending an approach based on a non-reversible diffusion. One idea is that the diffusion based on the drift {∇ log π(x) + γ(x)} is associated with the target π provided

∇ . {π(x)γ(x)} = 0

which holds for the Langevin diffusion when γ(x)=0, but produces a non-reversible process in the alternative. The Langevin choice γ(x)=0 happens to be the worst possible when considering the asymptotic variance. In practice however the diffusion need be discretised, which induces an approximation that may be catastrophic for convergence if not corrected, and a relapse into reversibility if corrected by Metropolis. The proposal in the paper is to use a Lie-Trotter splitting I had never heard of before to split between reversible [∇ log π(x)] and non-reversible [γ(x)] parts of the process. The deterministic part is chosen as γ(x)=∇ log π(x) [but then what is the point since this is Langevin?] or as the gradient of a power of π(x). Although I was mostly lost by that stage, the paper then considers the error induced by a numerical integrator related with this deterministic part, towards deriving asymptotic mean and variance for the splitting scheme. On the unit hypercube. Although the paper includes a numerical example for the warped normal target, I find it hard to visualise the implementation of this scheme. Having obviously not heeded Nicolas’ and James’ advice, the authors also analyse the Pima Indian dataset by a logistic regression!)

February 6, 2017 at 1:03 am

I think the authors might have intended γ(x)=J∇ log π(x), where J is an antisymmetric matrix, as the second step.

Visualise it as half a step of MALA (or any other metropolised scheme), and half a step moving quickly on level curves of the posterior.

Of course, there is always bias, which is a non-starter for many applications, but for models where you cannot even aspire to reach equilibrium within a reasonable amount of time, this method provides an approach to getting a reasonable estimate for an average (with a bias that can be somewhat quantified).