Non‐reversible parallel tempering: A scalable highly parallel MCMC scheme

Parallel tempering (PT) methods are a popular class of Markov chain Monte Carlo schemes used to sample complex high‐dimensional probability distributions. They rely on a collection of N interacting auxiliary chains targeting tempered versions of the target distribution to improve the exploration of the state space. We provide here a new perspective on these highly parallel algorithms and their tuning by identifying and formalizing a sharp divide in the behaviour and performance of reversible versus non‐reversible PT schemes. We show theoretically and empirically that a class of non‐reversible PT methods dominates its reversible counterparts and identify distinct scaling limits for the non‐reversible and reversible schemes, the former being a piecewise‐deterministic Markov process and the latter a diffusion. These results are exploited to identify the optimal annealing schedule for non‐reversible PT and to develop an iterative scheme approximating this schedule. We provide a wide range of numerical examples supporting our theoretical and methodological contributions. The proposed methodology is applicable to sample from a distribution π with a density L with respect to a reference distribution π0 and compute the normalizing constant ∫Ldπ0. A typical use case is when π0 is a prior distribution, L a likelihood function and π the corresponding posterior distribution.