We introduce a framework rooted in a rate distortion problem for Markov chains, and show how a suite of commonly used Markov Chain Monte Carlo (MCMC) algorithms are specific instances within it, where the target stationary distribution is controlled by the distortion function. Our approach offers a unified variational view on the optimality of algorithms such as Metropolis-Hastings, Glauber dynamics, the swapping algorithm and Feynman-Kac path models. Along the way, we analyze factorizability and geometry of multivariate Markov chains. Specifically, we demonstrate that induced chains on factors of a product space can be regarded as information projections with respect to a particular divergence. This perspective yields Han--Shearer type inequalities for Markov chains as well as applications in the context of large deviations and mixing time comparison.

翻译：我们引入了一个基于马尔可夫链率失真问题的框架，并展示了多种常用的马尔可夫链蒙特卡洛（MCMC）算法如何作为该框架的特例，其中目标平稳分布受失真函数控制。我们的方法为Metropolis-Hastings算法、Glauber动力学、交换算法以及Feynman-Kac路径模型等算法的最优性提供了统一的变分视角。在此过程中，我们分析了多变量马尔可夫链的可分解性与几何结构。具体而言，我们证明了乘积空间因子上的诱导链可视为关于特定散度的信息投影。这一视角不仅导出了马尔可夫链的Han--Shearer型不等式，还在大偏差理论和混合时间比较中产生了应用。