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On the rate of convergence of the Gibbs sampler for the 1-D Ising model by geometric bound

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  • Shiu, Shang-Ying
  • Chen, Ting-Li

Abstract

The second largest eigenvalue in absolute value determines the rate of convergence of the Markov chain Monte Carlo methods. In this paper we consider the Gibbs sampler for the 1-D Ising model. We apply the geometric bound by Diaconis and Stroock (1991) to calculate an upper bound of the second largest eigenvalue, which we show is also a bound of the second largest eigenvalue in absolute value. Based on this upper bound, we derive that the convergence time is O(n2), where n is the number of sites. Our result includes a constant of proportionality, which enables us to give a precise bound of the convergence time. The results presented in this paper provide the lowest bound compared to those with a constant of proportionality in the literature.

Suggested Citation

  • Shiu, Shang-Ying & Chen, Ting-Li, 2015. "On the rate of convergence of the Gibbs sampler for the 1-D Ising model by geometric bound," Statistics & Probability Letters, Elsevier, vol. 105(C), pages 14-19.
    • Handle: RePEc:eee:stapro:v:105:y:2015:i:c:p:14-19
    DOI: 10.1016/j.spl.2015.06.004
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    References listed on IDEAS

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    1. Chen, Ting-Li & Hwang, Chii-Ruey, 2013. "Accelerating reversible Markov chains," Statistics & Probability Letters, Elsevier, vol. 83(9), pages 1956-1962.
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    Cited by:

    1. Helali, Amine, 2019. "The convergence rate of the Gibbs sampler for generalized 1-D Ising model," Statistics & Probability Letters, Elsevier, vol. 154(C), pages 1-1.

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