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Quantitative convergence assessment for Markov chain Monte Carlo via cusums

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Abstract

Yu (1995) provides a novel convergence diagnostic for Markov chain Monte Carlo (MCMC) which provides a qualitative measure of mixing for Markov chains via a cusum path plot for univariate parameters of interest. The method is based upon the output of a single replication of an MCMC sampler and is therefore widely applicable and simple to use. One criticism of the method is that it is subjective in its interpretation, since it is based upon a graphical comparison of two cusum path plots. In this paper, we develop a quantitative measure of smoothness which we can associate with any given cusum path, and show how we can use this measure to obtain a quantitative measure of mixing. In particular, we derive the large sample distribution of this smoothness measure, so that objective inference is possible. In addition, we show how this quantitative measure may also be used to provide an estimate of the burn-in length for any given sampler. We discuss the utility of this quantitative approach, and highlight a problem which may occur if the chain is able to remain in any one state for some period of time. We provide a more general implementation of the method to overcome the problem in such cases.

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References

  • Brooks, S. P., Dellaportas, P. and Roberts, G. O. (1997) A total variation method for diagnosing convergence of MCMC algorithms. Journal of Computational and Graphical Statistics, 6, 251–65.

    Google Scholar 

  • Brooks, S. P. and Roberts, G. O. (1996) Diagnosing convergence of Markov chain Monte Carlo algorithms. Technical report, University of Cambridge.

  • Carlin, B. P. and Chib, S. (1995) Bayesian model choice via Markov chain Monte Carlo. Journal of the Royal Statistical Society, Series B, 57, 473–84.

    Google Scholar 

  • Cowles, M. K. and Carlin, B. P. (1996) Markov Chain Monte Carlo convergence diagnostics: a comparative review. Journal of the American Statistical Association, 91, 883–904.

    Google Scholar 

  • Gafarian, A. V., Ancker, C. J. and Morisaku, T. (1978) Evaluation of commonly used rules for detecting 'steady state' in computer simulation. Naval Research Logistics Quarterly, 25, 511–29.

    Google Scholar 

  • Gelfand, A. E. and Smith, A. F. M. (1990) Sampling based approaches to calculating marginal densities. Journal of the American Statistical Association, 85, 398–409.

    Google Scholar 

  • Gelman, A. and Rubin, D. (1992a) Inference from iterative simulation using multiple sequences. Statistical Science, 7, 457–511.

    Google Scholar 

  • Gelman, A. and Rubin, D. B. (1992b) A single series from the Gibbs sampler provides a false sense of security. In Bayesian Statistics 4, Bernardo, J. M., Berger, J. O., Dawid, A. P. and Smith, A. F. M. (eds), Oxford University Press, New York. pp. 625–31.

    Google Scholar 

  • Lin, Z. Y. (1992) On the increments of partial sums of a ø-mixing sequence. Theory of Probability and its Applications, 36, 316–26.

    Google Scholar 

  • Liu, C., Liu, J. and Rubin, D. B. (1993) A control variable for assessment the convergence of the Gibbs sampler. In: Proceedings of the Statistical Computing Section of the American Statistical Association, pp. 74–8.

  • Raftery, A. E. and Lewis, S. M. (1992) How many iterations in the Gibbs sampler? In Bayesian Statistics 4, Bernardo, J. M., Smith, A. F. M., Dawid, A. P. and Berger, J. O. (eds), Oxford University Press, New York, pp. 763–774

    Google Scholar 

  • Roberts, G. O. (1994) Methods for estimating L 2 convergence of Markov chain Monte Carlo. In Bayesian Statistics and Econometrics: Essays in Honor of Arnold Zellner, Berry, D., Chaloner, K. and Geweke, J. (eds), North Holland, Amsterdam.

    Google Scholar 

  • Williams, E. J. (1959) Regression Analysis. John Wiley and Sons

  • Yu, B. (1995) Discussion to Besag et al. (1995). Statistical Science, 10, 3–66.

    Google Scholar 

  • Yu, B. and Mykland, P. (1998) Looking at Markov sampler through cusum path plots: a simple diagnostic idea. Statistics and Computing, to appear.

Download references

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  1. School of Mathematics, University of Bristol University Walk, Bristol, BS8 1TW, UK

    S. P. Brooks

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  1. S. P. Brooks
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Brooks, S.P. Quantitative convergence assessment for Markov chain Monte Carlo via cusums. Statistics and Computing 8, 267–274 (1998). https://doi.org/10.1023/A:1008965613031

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