Adaptive Markov chain Monte Carlo (MCMC) algorithms, which automatically tune their parameters based on past samples, have proved extremely useful in practice. The self-tuning mechanism makes them `non-Markovian', which means that their validity cannot be ensured by standard Markov chains theory. Several different techniques have been suggested to analyse their theoretical properties, many of which are technically involved. The technical nature of the theory may make the methods unnecessarily unappealing. We discuss one technique -- based on a martingale decomposition -- with uniformly ergodic Markov transitions. We provide an accessible and self-contained treatment in this setting, and give detailed proofs of the results discussed in the paper, which only require basic understanding of martingale theory and general state space Markov chain concepts. We illustrate how our conditions can accomodate different types of adaptation schemes, and can give useful insight to the requirements which ensure their validity.

翻译：自适应马尔可夫链蒙特卡洛（MCMC）算法能够根据历史样本自动调整其参数，在实践中已被证明极具实用价值。这种自调节机制使其具有“非马尔可夫性”，这意味着标准马尔可夫链理论无法确保其有效性。目前已有多种不同的技术被提出用于分析其理论性质，其中许多在技术上较为复杂。理论的复杂性可能使这些方法显得不必要的艰深。本文讨论一种基于鞅分解的技术——适用于具有一致遍历性的马尔可夫转移过程。我们在此框架下提供了易于理解且自成一体的论述，并给出了论文中讨论结果的详细证明，这些证明仅需对鞅理论和一般状态空间马尔可夫链概念有基本理解。我们通过实例说明所提条件如何适用于不同类型的自适应方案，并能为确保其有效性的要求提供有价值的见解。