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.

翻译：提升采样器构成一类马尔可夫链蒙特卡洛方法，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例子，源自针对全序状态空间（如整数或实数）的随机游走Metropolis算法。提升采样器的构建方式是将提议分布拆分为两部分：一部分沿递增方向，另一部分沿递减方向。该采样器会持续沿某一方向移动，直至遭遇拒绝，此时它将翻转方向。就渐近方差而言，无论目标分布如何，它在不增加额外计算成本的情况下均优于随机游走Metropolis算法。然而，其他研究表明，超出这一简单情形后，提升采样器并非总能优于其对应的Metropolis算法。在本文中，我们借助Tierney（1998）的经典工作，在一个涵盖广泛类型提升采样器的通用框架内进行分析。我们的研究发现，本质上，无论目标分布、方向诱导方式以及提升采样器所源自的算法类型（无论是Metropolis–Hastings算法、可逆跳跃算法等），其渐近方差至多不会增加超过2倍。这一结果表明，尽管通过提升采样器可能获得显著收益，但潜在的损失却非常有限。