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.

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] Adaptive algorithms and stochastic approximations. Springer-Verlag, Berlin (1990). Translated from the French by Stephen S. Wilson. | MR 1082341 | Zbl 0752.93073

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

and ,[3] Convergence and robustness of the Robbins-Monro algorithm truncated at randomly varying bounds. Stochastic Process. Appl. 27 (1988) 217-231. | Zbl 0632.62082

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

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

, and ,[6] Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B 39 (1977) 1-38. | Zbl 0364.62022

, and ,[7] 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

and ,[8] 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

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

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

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

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

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

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

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

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

,