AI 中文总结
本研究在强空间混合条件下，通过三种方法证明了Gibbs采样器扰动分析中的“平方根”现象，即一步误差约为$1/\sqrt{\tau}$时局部更新仍有效，并展示了其在MCMC算法调参中的应用。
AI 中文摘要
马尔可夫链扰动分析的基本问题是其转移核的微小变化如何影响其平稳分布。经典扰动界通常要求核误差远小于$1/\tau$，其中$\tau$是混合时间或松弛时间。尽管这一尺度对于一般马尔可夫链是尖锐的，我们研究了一种一般的“平方根”现象，即对于局部更新，大约$1/\sqrt{\tau}$量级的一步误差可能是足够的。我们在Lin, Liu和Smith (2025)中在强假设下证明了这种现象的一种形式。这里我们大幅削弱了这些假设，并通过三种不同的方法证明这种现象成立。首先，块分解适用于两个链的平稳测度具有结构假设的情形。其次，近似块更新论证将结果扩展到统计相关的马尔可夫链蒙特卡洛（MCMC）设置，其中精确后验及其相关采样器具有结构保证，但扰动后验没有。第三，我们对一类具有硬约束的模型使用直接计算，其中两个一般结果都不能直接适用。我们在三个MCMC设置中说明了这些结果，并展示了它们如何直接指导近似MCMC算法的调参。
英文摘要
The basic question in perturbation analysis of Markov chains is how small changes in their transition kernels affect their stationary distributions. Classical perturbation bounds typically require the kernel error to be much smaller than $1/τ$, where $τ$ is a mixing or relaxation time. Although this scaling is sharp for Markov chains in general, we investigate a general "square-rooting" phenomenon in which one-step errors of order roughly $1/\sqrtτ$ can be sufficient for local updates. We proved a form of this phenomenon in Lin, Liu and Smith (2025) under strong assumptions. Here we substantially weaken these assumptions, and prove this phenomenon occurs using three distinct approaches. First, block factorization applies under structural assumptions on the stationary measures of both chains. Second, approximate block-update arguments extend the result to statistically relevant Markov chain Monte Carlo (MCMC) settings, where structural guarantees are available for the exact posterior and its associated sampler, but not for the perturbed posterior. Third, we use direct calculations for a class of models with hard constraints where neither general result is directly available. We illustrate these results in three MCMC settings and show how they directly inform the tuning of approximate MCMC algorithms.
