Statistics and Computing

, Volume 11, Issue 1, pp 83–87 | Cite as

Bounding convergence rates for Markov chains: An example of the use of computer algebra

  • John E. Kolassa


Kolassa and Tanner (J. Am. Stat. Assoc. (1994) 89, 697–702) present the Gibbs-Skovgaard algorithm for approximate conditional inference. Kolassa (Ann Statist. (1999), 27, 129–142) gives conditions under which their Markov chain is known to converge. This paper calculates explicity bounds on convergence rates in terms calculable directly from chain transition operators. These results are useful in cases like those considered by Kolassa (1999).

Markov chain Monte Carlo computer algebra 


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  1. Kolassa J.E. and Tanner M.A. 1994. Approximate conditional inference in exponential families via the Gibbs sampler. Journal of the American Statistical Association 89: 697-702.Google Scholar
  2. Kolassa J.E. 1999. Convergence and accuracy of Gibbs sampling for conditional distributions in generalized linear models. Ann. Statist., 27: 129-142.Google Scholar
  3. Kolassa J.E. 2000. Explicit bounds for geometric convergence of Markov chains. Journal of Applied Probability, In press.Google Scholar
  4. Meyn S.P. and Tweedie R.L. 1994. Computable bounds for geometric convergence rates of Markov chains. Advances in Applied Probability 4: 981-1011.Google Scholar
  5. Nummelin E. 1984. General Irreducible Markov Chains and Non-Negative Operators. Cambridge University Press, New York.Google Scholar
  6. Rosenthal J.S. 1995a. Rates of convergence for Gibbs sampling for variance components models. Annals of Statistics 23: 740-761.Google Scholar
  7. Rosenthal J.S. 1995b. Minorization conditions and convergence rates for Markov chain Monte Carlo. Journal of the American Statistical Association 90: 558-566.Google Scholar
  8. Roberts G.O. and Polson N.G. 1994. On the geometric convergence of the Gibbs sampler. Journal of the Royal Statistical Society Series B 56: 377-384.Google Scholar
  9. Schervish M.J. and Carlin B.P. 1992. On the convergence of successive substitution sampling.Journal of Computational and Graphical Statistics 1: 111-127.Google Scholar
  10. Tanner M.A. 1996. Tools for Statistical Inference. Springer-Verlag, Heidelberg.Google Scholar
  11. Tierney L. 1994. Markov Chains for exploring posterior distributions. Annals of Statistics 22: 1701-1762.Google Scholar
  12. Wolfram Research. 1996. Mathematica 3.0. Wolfram Research, Champaign, IL.Google Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • John E. Kolassa
    • 1
  1. 1.Rutgers UniversityPiscataway

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