Marathon：一个用于分析马尔可夫链蒙特卡罗算法的开源软件库
Marathon: An open source software library for the analysis of Markov-Chain Monte Carlo algorithms
摘要：
在本文中，我们考虑马尔可夫链蒙特卡罗（MCMC）方法的随机抽样的组合对象。这种算法的运行时间取决于底层马尔可夫链的总混合时间，并且通常是未知的。对于某些马尔可夫链，这个总混合时间的上界存在，但太大，在实践中是适用的。我们试图回答这个问题，是否总混合时间接近其上限，或者如果有一个显着的差距，他们之间。在这样做的时候，我们提出了软件库马拉松，旨在支持MCMC的采样算法的分析。该库的主要应用是计算表示马尔可夫链结构的所谓状态图的属性。我们使用马拉松调查质量的几个边界方法对四个著名的马尔可夫链抽样完美匹配和二分图实现。在一组实验中，我们计算了大量输入实例的总混合时间及其几个界限。我们发现由著名的正则路径方法得到的上界比总混合时间大几个数量级，并且随着输入大小的增加而恶化。与此相反，谱界被发现是一个精确的近似的总混合时间。
英文摘要：
In this paper, we consider the Markov-Chain Monte Carlo (MCMC) approach for random sampling of combinatorial objects. The running time of such an algorithm depends on the total mixing time of the underlying Markov chain and is unknown in general. For some Markov chains, upper bounds on this total mixing time exist but are too large to be applicable in practice. We try to answer the question, whether the total mixing time is close to its upper bounds, or if there is a significant gap between them. In doing so, we present the software library marathon which is designed to support the analysis of MCMC based sampling algorithms. The main application of this library is to compute properties of so-called state graphs which represent the structure of Markov chains. We use marathon to investigate the quality of several bounding methods on four well-known Markov chains for sampling perfect matchings and bipartite graph realizations. In a set of experiments, we compute the total mixing time and several of its bounds for a large number of input instances. We find that the upper bound gained by the famous canonical path method is several magnitudes larger than the total mixing time and deteriorates with growing input size. In contrast, the spectral bound is found to be a precise approximation of the total mixing time.
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