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.
Mots-clés : EM algorithm, SAEM algorithm, stochastic approximation, MCMC algorithm, convolution model, change-points model
@article{PS_2004__8__115_0,
author = {Kuhn, Estelle and Lavielle, Marc},
title = {Coupling a stochastic approximation version of {EM} with an {MCMC} procedure},
journal = {ESAIM: Probability and Statistics},
pages = {115--131},
publisher = {EDP-Sciences},
volume = {8},
year = {2004},
doi = {10.1051/ps:2004007},
mrnumber = {2085610},
zbl = {1155.62420},
language = {en},
url = {http://www.numdam.org/articles/10.1051/ps:2004007/}
}
TY - JOUR AU - Kuhn, Estelle AU - Lavielle, Marc TI - Coupling a stochastic approximation version of EM with an MCMC procedure JO - ESAIM: Probability and Statistics PY - 2004 SP - 115 EP - 131 VL - 8 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/ps:2004007/ DO - 10.1051/ps:2004007 LA - en ID - PS_2004__8__115_0 ER -
%0 Journal Article %A Kuhn, Estelle %A Lavielle, Marc %T Coupling a stochastic approximation version of EM with an MCMC procedure %J ESAIM: Probability and Statistics %D 2004 %P 115-131 %V 8 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/ps:2004007/ %R 10.1051/ps:2004007 %G en %F PS_2004__8__115_0
Kuhn, Estelle; Lavielle, Marc. Coupling a stochastic approximation version of EM with an MCMC procedure. ESAIM: Probability and Statistics, Tome 8 (2004), pp. 115-131. doi : 10.1051/ps:2004007. http://www.numdam.org/articles/10.1051/ps:2004007/
[1] , and, Adaptive algorithms and stochastic approximations. Springer-Verlag, Berlin (1990). Translated from the French by Stephen S. Wilson. | MR | Zbl
[2] and, Les algorithmes stochastiques contournent-ils les pièges ? C. R. Acad. Sci. Paris Ser. I Math. 321 (1995) 335-338. | Zbl
[3] , and, Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds. Stochastic Process. Appl. 27 (1988) 217-231. | Zbl
[4] and, A simulated pseudo-maximum likelihood estimator for nonlinear mixed models. Comput. Statist. Data Anal. 39 (2002) 187-201. | Zbl
[5] , and, Convergence of a stochastic approximation version of the EM algorithm. Ann. Statist. 27 (1999) 94-128. | Zbl
[6] , and, Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B 39 (1977) 1-38. | Zbl
[7] and, 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). | Zbl
[8] and, Maximum likelihood estimation for spatial models by Markov chain Monte Carlo stochastic approximation. J. R. Stat. Soc. Ser. B 63 (2001) 339-355. | Zbl
[9] , A gradient algorithm locally equivalent to the EM algorithm. J. R. Stat. Soc. Ser. B 57 (1995) 425-437. | Zbl
[10] and, An application of MCMC methods to the multiple change-points problem. Signal Processing 81 (2001) 39-53. | Zbl
[11] and, A simulated annealing version of the EM algorithm for non-Gaussian deconvolution. Statist. Comput. 7 (1997) 229-236.
[12] and, Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika 80 (1993) 267-278. | Zbl
[13] and, Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 (1996) 101-121. | Zbl
[14] and, Markov chains and stochastic stability, Springer-Verlag London Ltd., London. Comm. Control Engrg. Ser. (1993). | MR | Zbl
[15] , On the convergence properties of the EM algorithm. Ann. Statist. 11 (1983) 95-103. | Zbl
[16] , On recursive estimation in incomplete data models. Statistics 34 (2000) 27-51 (English). | Zbl
Cité par Sources :
