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A View of the EM Algorithm that Justifies Incremental, Sparse, and Other Variants

Radford M. Neal
Department of Statistics and
Department of Computer Science
University of Toronto

Geoffrey E. Hinton
Department of Computer Science
University of Toronto


The EM algorithm performs maximum likelihood estimation for data in which some variables are unobserved. We present a function that resembles negative free energy and show that the M step maximizes this function with respect to the model parameters and the E step maximizes it with respect to the distribution over the unobserved variables. From this perspective, it is easy to justify an incremental variant of the EM algorithm in which the distribution for only one of the unobserved variables is recalculated in each E step. This variant is shown empirically to give faster convergence in a mixture estimation problem. A variant of the algorithm that exploits sparse conditional distributions is also described, and a wide range of other variant algorithms are also seen to be possible.

In M. I. Jordan (editor) Learning in Graphical Models, pp. 355-368, Dordrecht: Kluwer Academic Publishers (1998)

Download:  [postscript] [pdf]
This is a revision of an earlier draft from February 1993.

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