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Marathon: An Open Source Software Library for the Analysis of Markov-Chain Monte Carlo Algorithms.

Steffen Rechner1, Annabell Berger1,2

  • 1Institute of Computer Science, Martin Luther University Halle-Wittenberg, Halle (Saale), Germany.

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Summary

The marathon software library aids Markov-Chain Monte Carlo (MCMC) analysis. Spectral bounds accurately approximate Markov chain mixing times, unlike canonical path methods which often overestimate.

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Area of Science:

  • Computer Science
  • Mathematics
  • Algorithm Analysis

Background:

  • Markov-Chain Monte Carlo (MCMC) methods are crucial for sampling complex probability distributions.
  • Analyzing the efficiency of MCMC algorithms often involves understanding their mixing time.
  • State graphs are fundamental for representing the structure of Markov chains.

Purpose of the Study:

  • Introduce the 'marathon' software library for MCMC sampling algorithm analysis.
  • Evaluate the performance of different bounding methods for Markov chain mixing times.
  • Compare the accuracy of canonical path and spectral bounds.

Main Methods:

  • Utilized the 'marathon' software library to analyze Markov chains.
  • Computed total mixing time and various bounds for sampling perfect matchings and bipartite graphs.
  • Experimentally investigated the quality of bounding methods on well-known Markov chains.

Main Results:

  • The canonical path method's upper bound frequently exceeds the total mixing time by orders of magnitude.
  • The canonical path bound's accuracy degrades as input size increases.
  • Spectral bounds provide a precise approximation of the total mixing time.

Conclusions:

  • The 'marathon' library is a valuable tool for MCMC analysis.
  • Spectral bounds are superior to canonical path bounds for estimating Markov chain mixing times.
  • The efficiency of MCMC sampling can be better predicted using spectral methods.