ESAIM: PS, April 2008, Vol. 12, p. 308-326
DOI: 10.1051/ps:2007041

On EM algorithms and their proximal generalizations

Stéphane Chrétien¹ and Alfred O. Hero<sup>2

1</sup>  Université de Franche-Comté, Laboratoire de Mathématiques, UMR CNRS 6623, 16 route de Gray, 25030 Besançon, France; chretien@math.univ-fcomte.fr
²  Department of Electrical Engineering and Computer Science, 1301 Beal St., University of Michigan, Ann Arbor, MI 48109-2122, USA; hero@eecs.umich.edu

Received June 14, 2007. Revised June 27, 2007. Published online 8 May 2008

Abstract
In this paper, we analyze the celebrated EM algorithm from the point of view of proximal point algorithms. More precisely, we study a new type of generalization of the EM procedure introduced in [Chretien and Hero (1998)] and called Kullback-proximal algorithms. The proximal framework allows us to prove new results concerning the cluster points. An essential contribution is a detailed analysis of the case where some cluster points lie on the boundary of the parameter space.

Mathematics Subject Classification. 65C20, 65C60

Key words: Maximum Likelihood Estimation (MLE), EM algorithm, proximal point algorithm, Karush-Kuhn-Tucker condition, mixture densities, competing risks models


© EDP Sciences, SMAI 2008