# An asymptotic Peskun ordering and its application to lifted samplers

arXiv: https://arxiv.org/abs/2003.05492 · HTML: https://mlsanity.com/paper/2003.05492/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Mar 2020.

## Built on by (most cited first, 2 of 2)

- [Improving multiple-try Metropolis with local balancing](https://mlsanity.com/paper/2211.11613/) (2022)
- [A fast non-reversible sampler for Bayesian mixture models](https://mlsanity.com/paper/2510.03226/) (2025)

## Abstract (arXiv)

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. It is however also known for being a result that is notably difficult to establish. Indeed, one has to prove that the probability to reach a state y from a state x, using a sampler, is greater than or equal to the probability using the other sampler, and this must hold for all pairs (x, y) such that x neq y. We provide in this paper a weaker version that does not require an inequality between the probabilities for all these states: essentially, 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 in great generality yields a qualitative conclusion: they asymptotically perform better in certain situations (and we are able to identify them), but not necessarily in others (and the reasons why are made clear). A quantitative study in a specific context of graphical-model simulation is also conducted.