Coupling a stochastic approximation version of EM with an MCMC procedure
ESAIM: Probability and Statistics, Volume 8 (2004), p. 115-131

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.

DOI : https://doi.org/10.1051/ps:2004007
Classification:  62F10,  62L20,  65C40
Keywords: 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},
     publisher = {EDP-Sciences},
     volume = {8},
     year = {2004},
     pages = {115-131},
     doi = {10.1051/ps:2004007},
     zbl = {1155.62420},
     zbl = {pre02161878},
     mrnumber = {2085610},
     language = {en},
     url = {http://www.numdam.org/item/PS_2004__8__115_0}
}
Kuhn, Estelle; Lavielle, Marc. Coupling a stochastic approximation version of EM with an MCMC procedure. ESAIM: Probability and Statistics, Volume 8 (2004) pp. 115-131. doi : 10.1051/ps:2004007. http://www.numdam.org/item/PS_2004__8__115_0/

[1] A. Benveniste, M. Métivier and P. Priouret, Adaptive algorithms and stochastic approximations. Springer-Verlag, Berlin (1990). Translated from the French by Stephen S. Wilson. | MR 1082341 | Zbl 0752.93073

[2] O. Brandière and M. Duflo, Les algorithmes stochastiques contournent-ils les pièges ? C. R. Acad. Sci. Paris Ser. I Math. 321 (1995) 335-338. | Zbl 0841.68048

[3] H.F. Chen, G. Lei and A.J. Gao, Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds. Stochastic Process. Appl. 27 (1988) 217-231. | Zbl 0632.62082

[4] D. Concordet and O.G. Nunez, A simulated pseudo-maximum likelihood estimator for nonlinear mixed models. Comput. Statist. Data Anal. 39 (2002) 187-201. | Zbl 1132.62337

[5] B. Delyon, M. Lavielle and E. Moulines, Convergence of a stochastic approximation version of the EM algorithm. Ann. Statist. 27 (1999) 94-128. | Zbl 0932.62094

[6] A.P. Dempster, N.M. Laird and D.B. Rubin, Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B 39 (1977) 1-38. | Zbl 0364.62022

[7] M.G. Gu and F.H. Kong, 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 0898.62101

[8] M.G. Gu and H.-T. Zhu, Maximum likelihood estimation for spatial models by Markov chain Monte Carlo stochastic approximation. J. R. Stat. Soc. Ser. B 63 (2001) 339-355. | Zbl 0979.62060

[9] K. Lange, A gradient algorithm locally equivalent to the EM algorithm. J. R. Stat. Soc. Ser. B 57 (1995) 425-437. | Zbl 0813.62021

[10] M. Lavielle and E. Lebarbier, An application of MCMC methods to the multiple change-points problem. Signal Processing 81 (2001) 39-53. | Zbl 1098.94557

[11] M. Lavielle and E. Moulines, A simulated annealing version of the EM algorithm for non-Gaussian deconvolution. Statist. Comput. 7 (1997) 229-236.

[12] X.-L. Meng and D.B. Rubin, Maximum likelihood estimation via the ECM algorithm: a general framework. Biometrika 80 (1993) 267-278. | Zbl 0778.62022

[13] K.L. Mengersen and R.L. Tweedie, Rates of convergence of the Hastings and Metropolis algorithms. Ann. Statist. 24 (1996) 101-121. | Zbl 0854.60065

[14] S.P. Meyn and R.L. Tweedie, Markov chains and stochastic stability, Springer-Verlag London Ltd., London. Comm. Control Engrg. Ser. (1993). | MR 1287609 | Zbl 0925.60001

[15] C.-F. Jeff Wu, On the convergence properties of the EM algorithm. Ann. Statist. 11 (1983) 95-103. | Zbl 0517.62035

[16] J.-F. Yao, On recursive estimation in incomplete data models. Statistics 34 (2000) 27-51 (English). | Zbl 0977.62092