Convergence of a Particle-based Approximation of the Block Online Expectation Maximization Algorithm

Online variants of the Expectation Maximization (EM) algorithm have recently been proposed to perform parameter inference with large data sets or data streams, in independent latent models and in hidden Markov models. Nevertheless, the convergence properties of these algorithms remain an open problem at least in the hidden Markov case. This contribution deals with a new online EM algorithm which updates the parameter at some deterministic times. Some convergence results have been derived even in general latent models such as hidden Markov models. These properties rely on the assumption that some intermediate quantities are available in closed form or can be approximated by Monte Carlo methods when the Monte Carlo error vanishes rapidly enough. In this paper, we propose an algorithm which approximates these quantities using Sequential Monte Carlo methods. The convergence of this algorithm and of an averaged version is established and their performance is illustrated through Monte Carlo experiments

On the Choice of the Number of Blocks with the Incremental EM Algorithm for the Fitting of Normal Mixtures

The EM algorithm is a popular method for parameter estimation in situations where the data can be viewed as being incomplete. As each E-step visits each data point on a given iteration, the EM algorithm requires considerable computation time in its application to large data sets. Two versions, the incremental EM (IEM) algorithm and a sparse version of the EM algorithm, were proposed recently by Neal R.M. and Hinton G.E. in Jordan M.I. (Ed.), Learning in Graphical Models, Kluwer, Dordrecht, 1998, pp. 355- 368 to reduce the computational cost of applying the EM algorithm. With the IEM algorithm, the available n observations are divided into B (B less than or equal to n) blocks and the E-step is implemented for only a block of observations at a time before the next M-step is performed. With the sparse version of the EM algorithm for the fitting of mixture models, only those posterior probabilities of component membership of the mixture that are above a specified threshold are updated; the remaining component-posterior probabilities are held fixed. In this paper, simulations are performed to assess the relative performances of the IEM algorithm with various number of blocks and the standard EM algorithm. In particular, we propose a simple rule for choosing the number of blocks with the IEM algorithm. For the IEM algorithm in the extreme case of one observation per block, we provide efficient updating formulas, which avoid the direct calculation of the inverses and determinants of the component-covariance matrices. Moreover, a sparse version of the IEM algorithm (SPIEM) is formulated by combining the sparse E-step of the EM algorithm and the partial E-step of the IEM algorithm. This SPIEM algorithm can further reduce the computation time of the IEM algorithm