Abstract
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 such that . 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.
Funding Statement
Philippe Gagnon acknowledges support from NSERC (Natural Sciences and Engineering Research Council of Canada) and FRQNT (Fonds de recherche du Québec – Nature et technologies). Florian Maire acknowledges support from NSERC.
Acknowledgements
The authors thank two anonymous referees for constructive comments that led to an improved manuscript.
Citation
Philippe Gagnon. Florian Maire. "An asymptotic Peskun ordering and its application to lifted samplers." Bernoulli 30 (3) 2301 - 2325, August 2024. https://doi.org/10.3150/23-BEJ1674
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