Markov chain Monte Carlo (MCMC) algorithms are based on the construction of a Markov Chain with transition probabilities $P_\mu(x,\cdot)$, where $\mu$ indicates an invariant distribution of interest. In this work, we look at these transition probabilities as functions of their invariant distributions, and we develop a notion of derivative in the invariant distribution of a MCMC kernel. We build around this concept a set of tools that we refer to as Markov Chain Monte Carlo Calculus. This allows us to compare Markov chains with different invariant distributions within a suitable class via what we refer to as mean value inequalities. We explain how MCMC Calculus provides a natural framework to study algorithms using an approximation of an invariant distribution, also illustrating how it suggests practical guidelines for MCMC algorithms efficiency. We conclude this work by showing how the tools developed can be applied to prove convergence of interacting and sequential MCMC algorithms, which arise in the context of particle filtering.

翻译：马尔可夫链蒙特卡洛（MCMC）算法基于构建转移概率为$P_\mu(x,\cdot)$的马尔可夫链，其中$\mu$表示感兴趣的不变分布。本研究将这些转移概率视为其不变分布的函数，并发展了关于MCMC核不变分布的导数概念。我们围绕这一概念构建了一套工具集，称为马尔可夫链蒙特卡洛微积分。通过所谓的均值不等式，我们得以比较适当类别内具有不同不变分布的马尔可夫链。我们阐释了MCMC微积分如何为研究使用近似不变分布的算法提供自然框架，同时说明了该框架如何为MCMC算法效率提出实用指导准则。最后，我们展示了所开发工具可用于证明粒子滤波背景下产生的交互式与序列化MCMC算法的收敛性。