Abstract
In this paper, we propose to monitor a Markov chain sampler using the cusum path plot of a chosen one-dimensional summary statistic. We argue that the cusum path plot can bring out, more effectively than the sequential plot, those aspects of a Markov sampler which tell the user how quickly or slowly the sampler is moving around in its sample space, in the direction of the summary statistic. The proposal is then illustrated in four examples which represent situations where the cusum path plot works well and not well. Moreover, a rigorous analysis is given for one of the examples. We conclude that the cusum path plot is an effective tool for convergence diagnostics of a Markov sampler and for comparing different Markov samplers.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brooks, S. P. (1996) Quantitative convergence diagnosis for MCMC via CUSUMS. To appear in Statistics and Computing.
Chan, K. S. and Tong, H. (1987) A note on embedding a discrete parameter ARMA model in a continuous parameter ARMA model. Journal of Time Series Analysis, 8, 277–281.
Chan, N. H. and Wei, C. Z. (1987) Asymptotic inference for nearly nonstationary AR(1) processes. Annals of Statistics, 15, 1050–63.
Cowles, M. K. and Carlin, B. P. (1996) Markov chain Monte Carlo convergence diagnostics: a comparative review. Journal of American Statistical Association, 91, 883–904.
Cui, L., Tanner, M. A., Sinhua, B. and Hall, W. J. (1992) Comment: monitoring convergence of the Gibbs sampler: further experience with the Gibbs stopper. Statistical Science, 7, 483–6.
Kotz, S. and Johnson, N. (1988). Encyclopedia of Statistical Sciences, Vol. 5, Wiley, New York.
Gelman, A., Roberts, G. O. and Gilks, W. R. (1996) Efficient Metropolis jumping rules, in Bayesian Statistics 5, Bernardo, J. M., Berger, J. O., Dawid, A. P. and Smith, A. F. M. (eds), Oxford University Press, Oxford, pp. 599–608.
Gelman, A. and Rubin, D. B. (1992a) 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, Oxford, pp. 625–32.
Gelman, A. and Rubin, D. B. (1992b) Inference from iterative simulation using multiple sequences. Statistical Science, 7, 457–511.
Gilks, W. R. and Roberts, G. O. (1993) Adaptive Markov chain Monte Carlo. IMS Bulletin, 22, 275.
Hastings, W. K. (1970) Monte-Carlo sampling methods using Markov chains and their applications. Biometrika, 57, 97–109.
Hall, P. and Heyde, C. C. (1980) Martingale Limit Theory and its Application. Academic Press, New York.
Lin, Z. Y. (1992) On the increments of partial sums of a ø-mixing sequence. Theoretical Probability and its Applications, 36, 316–26.
Mykland, P. A. (1994). Bartlett type identities for martingales. Annals of Statistics, 22, 21–38.
Philipp, W. and Stout, W. (1975) Almost sure invariance principles for partial sums of weakly dependent random variables, Memoirs of American Mathematical Society, Vol. 2(2), No. 161, Mamcau, Coden.
Rebolledo, R. (1980) Central limit theorems for local martingales. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete, 51, 269–86.
Robert, C. P. (1995) Convergence control methods for Markov chain Monte Carlo algorithms. Statistical Science, 10, 231–53
Yu, B. (1994) Estimating the L 1 error of kernel estimators based on Markov samplers. Technical Report 409, Statistics Department, University of California, Berkeley.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Yu, B., Mykland, P. Looking at Markov samplers through cusum path plots: a simple diagnostic idea. Statistics and Computing 8, 275–286 (1998). https://doi.org/10.1023/A:1008917713940
Issue Date:
DOI: https://doi.org/10.1023/A:1008917713940