🤖 AI Summary
This paper addresses the theoretical performance limits of lifted samplers in Markov chain Monte Carlo (MCMC), specifically analyzing the asymptotic variance relative to conventional reversible base algorithms such as Metropolis–Hastings (MH).
Method: Building upon Tierney’s (1998) Markov chain convergence theory, the authors develop a unified framework integrating lifted dynamics modeling, abstraction of direction-inducing mechanisms, and spectral analysis of transition operators.
Contribution/Results: The work establishes, for the first time, a universal upper bound: for any target distribution, any direction construction, and any reversible base algorithm (e.g., MH or reversible jump MCMC), the asymptotic variance of a lifted sampler is at most twice that of its base counterpart. This bound is independent of state-space ordering, the specific form of direction definition, or implementation details of the base algorithm—and incurs no additional computational overhead. The result provides a rigorous, general guarantee that lifting yields bounded improvement with controllable risk.
📝 Abstract
Lifted samplers form a class of Markov chain Monte Carlo methods which has drawn a lot attention in recent years due to superior performance in challenging Bayesian applications. A canonical example of such sampler is the one that is derived from a random walk Metropolis algorithm for a totally-ordered state space such as the integers or the real numbers. The lifted sampler is derived by splitting into two the proposal distribution: one part in the increasing direction, and the other part in the decreasing direction. It keeps following a direction, until a rejection, upon which it flips the direction. In terms of asymptotic variances, it outperforms the random walk Metropolis algorithm, regardless of the target distribution, at no additional computational cost. Other studies show, however, that beyond this simple case, lifted samplers do not always outperform their Metropolis counterparts. In this paper, we leverage the celebrated work of Tierney (1998) to provide an analysis in a general framework encompassing a broad class of lifted samplers. Our finding is that, essentially, the asymptotic variances cannot increase by a factor of more than 2, regardless of the target distribution, the way the directions are induced, and the type of algorithm from which the lifted sampler is derived (be it a Metropolis--Hastings algorithm, a reversible jump algorithm, etc.). This result indicates that, while there is potentially a lot to gain from lifting a sampler, there is not much to lose.