## non-reversibility in discrete spaces

**F**ollowing a recent JASA paper by Giacomo Zanella (which I have not yet read but is discussed on this blog), Sam Power and Jacob Goldman have recently arXived a paper on Accelerated sampling on discrete spaces with non-reversible Markov processes, where they use continuous-time, non-reversible algorithms à la PDMP, even though differential equations do not exist on discrete spaces. More specifically, they devise discrete versions of the coordinate sampler and of the Zig-Zag sampler, using Markov jump processes instead of differential equations, with detailed balance on the jump rate rather than the Markov kernel. A use of jump processes originating at least from Peskun (1973) and connected with MCMC algorithms in Matthew Stephens‘ 1999 PhD thesis. A neat thing about discrete settings is that the jump process can be implemented with no discretisation! However, as we noticed when working on birth-and-death processes with Olivier Cappé and Tobias Rydèn, there is a potential for disastrous implementation if an infinite sequence of instantaneous moves (out of zero probability states) is proposed.

The authors make the further assumption(s) that the discrete space is endowed with a graphical structure with a group G acting upon this graph, with an involution keeping the target (or a completion of the original target) invariant. In this framework, reversibility amounts to repeatedly using (group) generators þ with a low order (as in Bayesian variable selection, binary spin systems, where þ.þ=id, and other permutation problems), since they bring the chain back to its starting point. Their first sampler is called a Tabu sampler for avoiding such behaviour, forcing the next step to use other generators þ in the generator set Þ thanks to a binary auxiliary variable that partitions Þ into forward vs backward moves. For high order generators, the discrete coordinate and Zig-Zag samplers are instead repeatedly using the same generator (although it is unclear to me why this is beneficial, given that neither graph nor generator is not necessarily linked with the target). With the coordinate sampler being again much cheaper since it only looks at one direction in the generator group.

The paper contains a range of comparisons with (only) Zanella’s sampler, some presenting heavy gains in terms of ESS. Including one on hundreds of sensors in a football stadium. As I am not particularly familiar with these examples, except for the Bayesian variable selection one, I found it rather hard to determine whether or not the compared samplers were indeed exploring the entirety of the (highly complex and highly dimensional) target. The collection of examples is however quite rich and support the use of such non-reversible schemes. It may also be that the discrete nature of the target could facilitate the theoretical study of their convergence properties.

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