Abstract
This paper deals with a Bayesian analysis of a finite Beta mixture model. We present approximation method to evaluate the posterior distribution and Bayes estimators by Gibbs sampling, relying on the missing data structure of the mixture model. Experimental results concern contextual and non-contextual evaluations. The non-contextual evaluation is based on synthetic histograms, while the contextual one model the class-conditional densities of pattern-recognition data sets. The Beta mixture is also applied to estimate the parameters of SAR images histograms.
Similar content being viewed by others
References
McLachlan G. J. and Peel D. 2000. Finite Mixture Models. New York: Wiley.
Everitt B. S. and Hand D. J. 1981. Finite Mixture Distributions. Chapman and Hall, London, UK.
Roberts S. J. and Rezek L. 1998. Bayesian Approach to Gaussian Mixture Modeling. IEEE Transactions on Pattern Analysis and Machine Intelligence 20(11): 1133–1142.
Samuel K., Ng K. W., and Fang K. 1990. Symmetric Multivariate and Related Distributions. London/New York: Chapman and Hall.
Bouguila N., Ziou D., and Vaillancourt J. November 2004. Unsupervised Learning of a Finite Mixture Model Based on the Dirichlet Distribution and its Application. IEEE Transactions on Image Processing 13(11): 1533–1543.
Beckman R. J. and Tietjen G. L. 1978. Maximum Likelihood Estimation for the Beta Distribution. Journal of Statistics and Computational Simulation 7: 253–258.
Klieter G. 1992. Bayesian Diagnosis in Expert Systems. In AIJ92.
Lee J. C. and Lio Y. L. 1999. A Note on Bayesian Estimation and Prediction for the Beta-binomial Model. Journal of Statistical Computation and Simulation (63): 73–91.
Dempster A. P., Laird N. M., and Rubin D. B. 1977. Maximum Likelihood from Incomplete Data via the EM Algorithm. Journal of the Royal Statistical Society, B 39: 1–38.
McLachlan G. J. and Krishnan T. 1997. The EM Algorithm and Extensions. New York: Wiley.
Robert C. P. and Casella G. 1999. Monte Carlo Statistical Methods. Springer-Verlag.
Diebolt J. and Robert C. P. 1994. Estimation of Finite Mixture Distributions Through Bayesian Sampling. Journal of the Royal Statistical Society, B 56(2): 363–375.
Tsung I. L., Jack C. L., and Huey F. N. 2004. Bayesian Analysis of Mixture Modeling Using the Multivariate t Distribution. Statistics and Computing 14: 119–130.
Tsionas E. G. 2004. Bayesian Inference for Multivariate Gamma Distributions. Statistics and Computing 14: 223–233.
Celeux G. and Diebolt J. 1985. The SEM Algorithm: a Probabilistic Teacher Algorithm Derived from the EM Algorithm for the Mixture Problem. Computational Statistics Quarterly 2(1): 73–82.
Celeux G. and Diebolt J. 1992. A Stochastic Approximation Type EM Algorithm for the Mixture Problem. Stochastics and Stochastics Reports 41: 119–134.
Escobar M. and West M. 1995. Bayesian Prediction and Density Estimation. Journal of the American Statistical Association 90: 577–588.
Marin J. M., Mengersen K., and Robert C. P. 2004. Bayesian modeling and inference on mixtures of distributions. In D. Dey and C.R. Rao, editors, Handbook of Statistics 25. Elsevier-Sciences.
Robert C. P. and Rousseau J. 2002. A Mixture Approach To Bayesian Goodness of Fit. Technical Report 02009, Cahier du CEREMADE, Université Paris Dauphine.
Casella G., Mengersen K., Robert C., and Titterington D. 2000. Perfect Slice Samplers for Mixtures of Distributions. Journal of the Royal Statistical Society, B 64(4): 777–790.
Bezdek J. C. 1981. Pattern Recognition with Fuzzy Objective Function Algorithms. Plenum Press, New York.
Fielitz B. D and Myers B. L 1975. Estimation of Parameters in the Beta Distribution. Decision Sciences 6: 1–13, 1975.
Crawford S. L. 1994. An application of the Laplace Method to Finite Mixture Distributions. Journal of the American Statistical Association 89: 259–267.
Schwarz G. 1978. Estimating the Dimension of a Model. Annals of Statistics 6: 461–464.
Biernacki C., Celeux G. and Govaert G. 2000. Assessing a Mixture Model for Clustering with the Integrated Complete Likelihood. IEEE Transactions on Pattern Analysis and Machine Intelligence 22(7): 719–725.
Kass R. E. and Raftery A. E. 1995. Bayes Factor. Journal of the American Statistical Association 90: 733–795.
Roeder K. and Wasserman L. 1997. Practical Bayesian Density Estimation Using Mixture of Normals. Journal of the American Statistical Association 92: 894–902.
Richardson S. and Green P. J. 1997. On Bayesian Analysis of Mixtures with an Unknown Number of Components (With Discussion). Journal of the Royal Statistical Society, B 59: 731–792.
Stephens M. 2000. Bayesian Analysis of mixture Models with an Unknown Number of Components: An Alternative to reversible Jump Methods. Annals of Statistics 28:40–74.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bouguila, N., Ziou, D. & Monga, E. Practical Bayesian estimation of a finite beta mixture through gibbs sampling and its applications. Stat Comput 16, 215–225 (2006). https://doi.org/10.1007/s11222-006-8451-7
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11222-006-8451-7