论文标题
关于马尔可夫链收敛分析中单步漂移和较小化的局限性
On the limitations of single-step drift and minorization in Markov chain convergence analysis
论文作者
论文摘要
在过去的三十年中，在应用概率社区中竭尽全力开发界定一般状态空间马尔可夫链的收敛速率的技术。这些结果中的大多数假定存在漂移和较小（D \＆M）条件。经常观察到，基于单步d \＆M的收敛速率范围往往过于保守，尤其是在高维情况下。本文建立了研究这种现象的框架。结果表明，基于一组D \＆M条件绑定的任何收敛速率都不能比某个未知的最佳结合更好。策略旨在将界限放在最佳界限上，这允许人们量化基于D \＆M的收敛速率界限的程度。新理论应用于几个示例，包括高斯自回归过程（其真正的收敛率）和大都会调整后的langevin算法。结果强烈表明，基于单步d \＆M条件的收敛速率界限在高维度的情况下完全不足。
Over the last three decades, there has been a considerable effort within the applied probability community to develop techniques for bounding the convergence rates of general state space Markov chains. Most of these results assume the existence of drift and minorization (d\&m) conditions. It has often been observed that convergence rate bounds based on single-step d\&m tend to be overly conservative, especially in high-dimensional situations. This article builds a framework for studying this phenomenon. It is shown that any convergence rate bound based on a set of d\&m conditions cannot do better than a certain unknown optimal bound. Strategies are designed to put bounds on the optimal bound itself, and this allows one to quantify the extent to which a d\&m-based convergence rate bound can be sharp. The new theory is applied to several examples, including a Gaussian autoregressive process (whose true convergence rate is known), and a Metropolis adjusted Langevin algorithm. The results strongly suggest that convergence rate bounds based on single-step d\&m conditions are quite inadequate in high-dimensional settings.
