Summary
Many papers (including most of the papers in this issue of Computational Statistics) deal with Markov Chain Monte Carlo (MCMC) methods. This paper will give an introduction to the augmented Gibbs sampler (a special case of MCMC), illustrated using the random intercept model. A’ nonstandard’ application of the augmented Gibbs sampler will be discussed to give an illustration of the power of MCMC methods. Furthermore, it will be illustrated that the posterior sample resulting from an application of MCMC can be used for more than determination of convergence and the computation of simple estimators like the a posteriori expectation and standard deviation. Posterior samples give access to many other inferential possibilities. Using a simulation study, the frequency properties of some of these possibilities will be evaluated.
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Box, G.E.P. and Tiao, C. (1973). Bayesian Inference in Statistical Analysis. London: Addison-Wesley.
Browne, W.J. (1998). Applying MCMC Methods to Multi-level Models. Unpublished doctoral dissertation, University of Bath, United Kingdom.
Bryk, A.S. and Raudenbush S.W. (1992). Hierarchical Linear Models: Applications and Data Analysis Methods. London: SAGE. tem
Carlin, B.P. and Louis, T.A. (1996). Bayes and Empirical Bayes Methods for Data Analysis. London: Chapman and Hall.
Casella, G. and George, E. (1992). Explaining the Gibbs sampler. The American Statistician, 46, 167–174.
Chib, S. and Greenberg, E. (1995). Understanding the Metropolis-Hastings Algorithm. The American Statistician, 49, 327–335.
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.
Dempster, A.P., Laird, N.M. and Rubin, D.B. (1977). Maximum likelihood estimation from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B, 39, 1–38.
DiCiccio, T.J., Kass, R.E., Raftery, A. and Wasserman, L. (1997). Computing Bayes factors by combining simulation and asymptotic approximations. Journal of the American Statistical Association, 92, 903–915.
Gelfand, A.E., Hills, S.E., Racine-Poon, A. and Smith, A.F.M. (1990). Illustration of Bayesian inference in normal data models using Gibbs sampling. Journal of the American Statistical Association, 85, 972–985.
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.
Gelman, A., Carlin, J.B., Stern, H.S. and Rubin, D.B. (1995). Bayesian Data Analysis. Chapman and Hall, London.
Gelman, A., Meng, X. and Stern, H.S. (1996). Posterior predictive assessment of model fitness via realized discrepancies (with discussion). Statistica Sinica, 6, 733–760.
Goldstein, H. (1987). Multilevel Models in Educational and Social Research. London: Charles Griffin.
Goldstein, H. and Spiegelhalter, D.J. (1996). League tables and their limitations: statistical issues in comparisons of institutional performance (with discussion). Journal of the Royal Statistical Society, Series A, 159, 385–444.
Hobert, J.P. and Casella, G. (1996). The effect of improper priors on Gibbs sampling in hierarchical linear mixed models. Journal of the American Statistical Association, 91, 1461–1473.
Hoijtink, H. (1998). Constrained latent class analysis using the Gibbs sampler and posterior predictive p-values: applications to educational testing. Statistica Sinica, 8, 691–711.
Kass, R.E. and Raftery, A.E. (1995). Bayes factors. Journal of the American Statistical Association, 90, 773–795.
Meng, X.L. (1994). Posterior Predictive p-Values. The Annals of Statistics, 22, 1142–1160.
Mortimore, P., Sammons, P., Stoll, L., Lewis, D. and Ecob, R. (1988). School Matters, the Junior Years. Wells, Open Books.
Multilevel Models Project (1991). ML3 Data Library. London: Multilevel Models Project, Institute of Education, University of London.
Rubin, D.B. (1984). Bayesian justifiable and relevant frequency calculations for the applied statistician. The Annals of Statistics, 12, 1151–1172.
Self, S.G. and Liang, K.Y. (1987). Asymptotic properties of maximum likelihood estimators and likelihood ratio tests under nonstandard conditions. Journal of the American Statistical Association, 82, 605–610.
Smith, A.F.M. and Roberts, G.O. (1993). Bayesian computation via the Gibbs sampler and related Markov Chain Monte Carlo methods. Journal of the Royal Statistical Society, Series B, 55, 3–24.
Spiegelhalter, D., Thomas, A., Best, N. and Gilks, W. (1996a). BUGS 0.5 Bayesian Inference Using Gibbs Sampling. Manual (version ii).
Spiegelhalter, D., Thomas, A., Best, N. and Gilks, W. (1996b). BUGS 0.5 Examples. Volume 1 (version i).
Tanner, M.A. (1993). Tools for Statistical Inference. Springer, New York.
Tanner, M.A. and Wong, W.H. (1987). The calculation of posterior distributions by data augmentation. Journal of the American Statistical Association, 82, 528–540.
Tierney, L. (1994). Markov chains for exploring posterior distributions (with discussion). Annals of Statistics, 22, 1701–1762.
Zeger, S.L. and Karim, M.R. (1991). Generalized linear models with random effect; a Gibbs sampling approach. Journal of the American Statistical Association, 86, 79–86.
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Hoijtink, H. Posterior inference in the random intercept model based on samples obtained with Markov chain Monte Carlo methods. Computational Statistics 15, 315–336 (2000). https://doi.org/10.1007/s001800000037
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DOI: https://doi.org/10.1007/s001800000037