Diebolt, J. and Robert, C. P., `Bayesian estimation of finite mixture distributions,' J. R. Statist. Soc. B, 56, 363--375, 1994.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....on k given the data and the models. Even for the relatively simple Gaussian mixture model, this posterior cannot be calculated in closed form and must either be approximated analytically or estimated via sampling techniques such as Markov Chain Monte Carlo method (MCMC) Lavine and West (1992) Diebolt and Robert (1994) and Bensmail et al. 1997) provide examples of the application of sampling techniques to Bayesian inference for mixture models. The Bayesian and penalized likelihood approaches can be viewed from a single perspective by noting that the penalized likelihood methods can each be derived as ....

Diebolt, J. and Robert, C. P., `Bayesian estimation of finite mixture distributions,' J. R. Statist. Soc. B, 56, 363--375, 1994.

....complexity of integrating over the parameter space to get the posterior probabilities on k. The AutoClass algorithm (Cheeseman and Stutz 1996) uses various approximations to get around the computational issues. Sampling techniques have also been applied to this problem with some success (cf. Diebolt and Robert 1994). A third method (related to the Bayesian approach, see Chickering and Heckerman, 1996) is that of penalized likelihood (such as the Bayesian Information Criterion (BIC) and various coding based (e.g. MDL MML) criteria) A penalty term is added to the log likelihood to penalize the number of ....

Diebolt, J. and Robert, C. P. 1994. `Bayesian estimation of finite mixture distributions,' J. R. Stat. Soc.

....i s. 3 Simulate the parameters of the model according to their posterior distributions conditional on the i s. The validity of this procedure, namely the fact that the Markov chain associated with the algorithm converges in distribution to the true posterior distribution of , was shown by Diebolt and Robert (1990) in the context of one dimensional normal mixtures. Their proof is based on a duality principle, which uses the finite space nature of the chain associated with the i s. This chain is ergodic with state space f1; Kg and is thus geometrically convergent and even mixing. These properties ....

Diebolt, J., Robert, C.P. (1990), " Bayesian estimation of finite mixture distributions Part II: sampling implementation," Rapport Technique no. 110, LSTA, Universit'e Paris VI.

....the Bayesian analysis of mixtures. For instance, Lavine and West (1992) and Soubiran, Celeux, Diebolt, and Robert (1991) have used the Gibbs Sampler for estimating the parameters of a multivariate Gaussian mixture without assuming any specific characteristics for the component variance matrices. Diebolt and Robert (1994) have considered the Gibbs sampler and the Data Augmentation method of Tanner and Wong (1987) for general univariate Gaussian mixtures and proved that both algorithms converge in distribution to the true posterior distribution of the mixture parameters. Like these authors, we use conjugate priors ....

....chain is ergodic with state space f1; Kg and is thus geometrically convergent and even mixing. These properties transfer automatically to the sequence of values of and , and important properties such the Central Limit Theorem or the Law of the Iterated Logarithm are then satisfied (Diebolt and Robert 1994; Robert 1993) The same results also apply here, the only difference being the more complex simulation structure imposed by the variance assumptions. Steps 1 and 2 do not depend on the considered model. Step 1 is straightforward, and Step 2 consists of simulating from its conditional posterior ....

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Diebolt, J., Robert, C.P. (1994, " Bayesian estimation of finite mixture distributions," Journal of the Royal Statistical Society B 56 363-375.

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Diebolt, J. and Robert, C. P., `Bayesian estimation of finite mixture distributions,' J. R. Statist. Soc. B, 56, 363--375, 1994.

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Diebolt, J. and Robert C. (1990). Bayesian estimation of finite mixture distributions, Part II: Sampling implementation. Technical Report , L.S.T.A., Universite Paris VI.

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