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 occurs, 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.

翻译：提升采样器作为一类马尔可夫链蒙特卡罗方法，近年来因其在挑战性贝叶斯应用中的卓越性能而备受关注。此类采样器的一个典型示例源自针对全序状态空间（如整数或实数）的随机游走Metropolis算法。提升采样器通过将提议分布拆分为两部分得到：一部分沿递增方向，另一部分沿递减方向。该采样器持续沿某一方向移动，直至发生拒绝事件，此时翻转方向。就渐近方差而言，无论目标分布如何，它在不增加计算成本的前提下均优于随机游走Metropolis算法。然而，其他研究表明，超越这一简单情形后，提升采样器并非总能优于其对应的Metropolis算法变体。本文基于Tierney（1998）的经典工作，在涵盖广泛提升采样器类别的通用框架下进行分析。我们的研究发现，无论目标分布、方向诱导方式及提升采样器所源自的算法类型（包括Metropolis-Hastings算法、可逆跳转算法等），其渐近方差至多不会增加超过2倍。这一结果表明，虽然提升采样器可能带来显著增益，但其潜在损失极为有限。