Abstract
This paper describes a Bayesian approach to mixture modelling and a method based on predictive distribution to determine the number of components in the mixtures. The implementation is done through the use of the Gibbs sampler. The method is described through the mixtures of normal and gamma distributions. Analysis is presented in one simulated and one real data example. The Bayesian results are then compared with the likelihood approach for the two examples.
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References
Berger, J. O. (1985) Statistical Decision Theory and Bayesian Analysis, 2nd edn. Springer-Verlag, New York.
Box, G. E. P. (1980) Sampling and Bayes' inference in scientific modeling and robustness (with discussion). Journal of the Royal Statistical Society, B, 143, 383–430.
Crawford, S. L. (1994) An application of the Laplace method to finite mixture distributions. Journal of the American Statistical Association, 89, 259–67.
Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977) Maximum likelihood estimation from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, Series B, 39, 1–38.
Diebolt, J. and Robert, C. (1991) Bayesian estimation of finite mixture distributions part II: sampling implementation. Unpublished Report, L. S. T. A., Université Paris VI.
Diebolt, J. and Robert, C. (1994) Estimation of finite mixture distributions through Bayesian sampling. Journal of the Royal Statistical Society, Series B, 56, 363–75.
Evans, M., Guttman, I. and Olkin, I. (1992) Numerical aspects in estimating the parameters of a mixture of normal distributions. Journal of Computational and Graphical Statistics, 1, 351–65.
Everitt, B. S. and Hand, D. J. (1981) Finite Mixture Distributions. Chapman and Hall, New York.
Geisser, S. (1982) Aspects of the predictive and estimative approaches in the determination of probabilities. Biometrics Supplement, 38, 75–85.
Geisser, S. and Eddy, W. (1979) A predictive approach to model selection. Journal of the American Statistical Association, 74, 153–60.
Gelfand, A. E. and Dey, D. K. (1994) Bayesian model choice: asymptotics and exact calculations. Journal of the Royal Statistical Society, Series B, 56, 501–14.
Gelfand, A. E., Dey, D. K. and Chang, H. (1992) Model determination using predictive distributions, with implementation via sampling-based methods. In Bayesian Statistics 4, J. M. Bernado, J. O. Berger, A. P. Dawid and A. F. M. Smith (eds.), pp. 147–69. Oxford University Press, Oxford.
Gelfand, A. E., Mallick, B. K. and Dey, D. K. (1995) Modeling expert opinion arising as a partial probabilistic specification. Journal of the American Statistical Association, 90, 398–409.
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. and Rubin, D. B. (1992) Inference from iterative simulation using multiple sequences (with discussion). Statistical Science, 7, 457–511.
Geman, S. and Geman, D. (1984) Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–41.
George, E. I., Makov, U. E. and Smith, A. F. M. (1993) Conjugate likelihood distributions. Scandinavian Journal of Statistics, 20, 147–56.
Geweke, J. (1989) Bayesian inference in econometric models using Monte Carlo integration. Econometrica, 57, 1317–39.
Gilks, W. R. and Wild, P. (1992) Adaptive rejection sampling for Gibbs sampling. Applied Statistics, 41, 337–48.
Jaisingh, L. R., Kolarik, W. J. and Dey, D. K. (1987) A flexible bathtub hazard model for non-repairable systems with uncensored data. Microelectronics and Reliability, 27, 87–103.
Jeffreys, H. (1961) Theory of Probability, 3rd edn. Oxford University Press.
McLachlan, G. J. (1987) On bootstrapping the likelihood ratio test statistic for the number of components in a normal mixture. Applied Statistics, 36, 318–24.
McLachlan, G. J. and Basford, K. E. (1988) Mixture Models: Inference and Applications to Clustering. Marcel Dekker, New York.
Nelson, W. (1982) Applied Life Data Analysis. John Wiley, New York.
Stone, M. (1974) Cross-validatory choice and assessment of statistical predictions (with discussion). Journal of the Royal Statistical Society, Series B, 36, 111–47.
Tierney, L. (1995) Markov chains for exploring posterior distributions. The Annals of Statistics, 22, 1701–1721.
Titterington, D. M., Smith, A. F. M. and Makov, U. E. (1985) Statistical Analysis of Finite Mixture Distributions. John Wiley, New York.
West, M. (1992) Modeling with mixtures. In Bayesian Statistics 4, J. M. Bernado, J. O. Berger, A. P. Dawid and A. F. M. Smith (eds), pp. 503–24. Oxford University Press.
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Dey, D.K., Kuo, L. & Sahu, S.K. A Bayesian predictive approach to determining the number of components in a mixture distribution. Stat Comput 5, 297–305 (1995). https://doi.org/10.1007/BF00162502
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DOI: https://doi.org/10.1007/BF00162502