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Coupling a stochastic approximation version of EM with an MCMC procedure

Published online by Cambridge University Press:  15 September 2004

Estelle Kuhn  and
Marc Lavielle
Estelle Kuhn
Affiliation:
Université Paris Sud, Bât. 425, 91400 Orsay, France; Estelle.Kuhn@math.u-psud.fr.
Marc Lavielle
Affiliation:
Université René Descartes and Université Paris Sud, Bât. 425, 91400 Orsay, France; Marc.Lavielle@math.u-psud.fr.
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Abstract

The stochastic approximation version of EM (SAEM) proposed by Delyon et al. (1999) is a powerful alternative to EM when the E-step is intractable. Convergence of SAEM toward a maximum of the observed likelihood is established when the unobserved data are simulated at each iteration under the conditional distribution. We show that this very restrictive assumption can be weakened. Indeed, the results of Benveniste et al. for stochastic approximation with Markovian perturbations are used to establish the convergence of SAEM when it is coupled with a Markov chain Monte-Carlo procedure. This result is very useful for many practical applications. Applications to the convolution model and the change-points model are presented to illustrate the proposed method.

Keywords

EM algorithmSAEM algorithmstochastic approximationMCMC algorithmconvolution modelchange-points model.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 8 , August 2004 , pp. 115 - 131
Copyright
© EDP Sciences, SMAI, 2004

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References

A. Benveniste, M. Métivier and P. Priouret, Adaptive algorithms and stochastic approximations. Springer-Verlag, Berlin (1990). Translated from the French by Stephen S. Wilson.
Brandière, O. and Duflo, M., Les algorithmes stochastiques contournent-ils les pièges ? C. R. Acad. Sci. Paris Ser. I Math. 321 (1995) 335–338.
Chen, H.F., Lei, G. and Gao, A.J., Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds. Stochastic Process. Appl. 27 (1988) 217–231. CrossRef
Concordet, D. and Nunez, O.G., A simulated pseudo-maximum likelihood estimator for nonlinear mixed models. Comput. Statist. Data Anal. 39 (2002) 187–201. CrossRef
Delyon, B., Lavielle, M. and Moulines, E., Convergence of a stochastic approximation version of the EM algorithm. Ann. Statist. 27 (1999) 94–128.
Dempster, A.P., Laird, N.M. and Rubin, D.B., Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B 39 (1977) 1–38.
Gu, M.G. and Kong, F.H., A stochastic approximation algorithm with Markov chain Monte-Carlo method for incomplete data estimation problems. Proc. Natl. Acad. Sci. USA 95 (1998) 7270–7274 (electronic). CrossRef
Gu, M.G. and Zhu, H.-T., Maximum likelihood estimation for spatial models by Markov chain Monte Carlo stochastic approximation. J. R. Stat. Soc. Ser. B 63 (2001) 339–355. CrossRef
Lange, K., A gradient algorithm locally equivalent to the EM algorithm. J. R. Stat. Soc. Ser. B 57 (1995) 425–437.
Lavielle, M. and Lebarbier, E., An application of MCMC methods to the multiple change-points problem. Signal Processing 81 (2001) 39–53. CrossRef
Lavielle, M. and Moulines, E., A simulated annealing version of the EM algorithm for non-Gaussian deconvolution. Statist. Comput. 7 (1997) 229–236. CrossRef
Meng, X.-L. and Rubin, D.B., Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika 80 (1993) 267–278. CrossRef
Mengersen, K.L. and Tweedie, R.L., Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 (1996) 101–121.
S.P. Meyn and R.L. Tweedie, Markov chains and stochastic stability, Springer-Verlag London Ltd., London. Comm. Control Engrg. Ser. (1993).
Jeff Wu, C.-F., On the convergence properties of the EM algorithm. Ann. Statist. 11 (1983) 95–103.
Yao, J.-F., On recursive estimation in incomplete data models. Statistics 34 (2000) 27–51 (English). CrossRef