Abstract
We present the software library marathon, which is designed to support the analysis of sampling algorithms that are based on the Markov-Chain Monte Carlo principle. The main application of this library is the computation of properties of so-called state graphs, which represent the structure of Markov chains. We demonstrate applications and the usefulness of marathon by investigating the quality of several bounding methods on four well-known Markov chains for sampling perfect matchings and bipartite graphs. 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 often 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.
Highlights
What happens if we stop the random walk after a finite number of steps? The number of steps which are necessary to sample from a probability distribution which is close to the desired distribution is known as the total mixing time of a Markov chain, and is of central interest for the applicability of an Markov-Chain Monte Carlo (MCMC) algorithm
Each input instance corresponds to two data points, showing the ratio of the upper spectral bound, and congestion bound with the total mixing time
We immediately observe that the spectral bound is very close to the total mixing time for all instances, in contrast to the congestion bound
Summary
Data Availability Statement: Source code is available at https://github.com/srechner/marathon. The task of random sampling is to return a randomly selected object from a typically large set of objects according to a specified probability distribution. Such tasks often arise in practical applications like network analysis, where properties of a certain network of interest are to be compared with those of a random null model network [1, 2]. Another application is approximate counting of combinatorial objects. An overview of the surprisingly versatile applications of the MCMC approach was given by Diaconis [4]
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