An asymptotic Peskun ordering and its application to lifted samplers

A Peskun ordering between two samplers, implying a dominance of one over the other, is known among the Markov chain Monte Carlo community for being a remarkably strong result, but it is also known for being one that is notably difficult to establish. Indeed, one has to prove that the probability to reach a state, using a sampler, is greater than or equal to the probability using the other sampler, and this must hold for all states excepting the current state. We provide in this paper a weaker version that does not require an inequality between the probabilities for all these states: the dominance holds asymptotically, as a varying parameter grows without bound, as long as the states for which the probabilities are greater than or equal to belong to a mass-concentrating set. The weak ordering turns out to be useful to compare lifted samplers for partially-ordered discrete state-spaces with their Metropolis-Hastings counterparts. An analysis yields a qualitative conclusion: they asymptotically perform better in certain situations (and we are able to identify these situations), but not necessarily in others (and the reasons why are made clear). The difference in performance is evaluated quantitatively in important applications such as graphical model simulation and variable selection. The code to reproduce all numerical experiments is available online.