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 lifted samplers 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：提升采样器是一类马尔可夫链蒙特卡洛方法，近年来因其在具有挑战性的贝叶斯应用中的优异性能而受到广泛关注。提升采样器的一个典型例子源于针对全序状态空间（例如整数或实数）的随机游走Metropolis算法。该提升采样器通过将提议分布分为两部分来构建：一部分对应递增方向，另一部分对应递减方向。它持续沿同一方向推进，直至遇到拒绝，此时方向翻转。就渐近方差而言，无论目标分布如何，该算法均在不增加计算成本的前提下优于随机游走Metropolis算法。然而，其他研究表明，在此简单情形之外，提升采样器并非总是优于对应的Metropolis算法。本文借助Tierney（1998）的经典工作，在一个涵盖广泛提升采样器类别的通用框架下进行分析。我们的发现是：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、可逆跳转算法等），渐近方差最多不会增加超过2倍。这一结果表明，尽管提升采样器可能带来显著收益，但其损失风险却极为有限。