The EM Algorithm and its Extensions

Since it is simple and stable, the EM algorithm (Dempster, Laird, and Rubin, 1977, JRSS-B) has been widely used to fit models from incomplete data. Our current research program in this area includes the following.

1. Acceleration

• The PX-EM algorithm (Chuanhai Liu, Donald B. Rubin, and Ying Nian Wu, 1998), shares the simplicity and stability of ordinary EM but is often much faster. The intuitive idea is to use a covariance adjustment to correct the M step, capitalizing on extra information captured in the imputed complete data. This is accomplished by parameter expansion; we expand the complete-data model while preserving the observed-data model and use the expanded complete-data model in the EM algorithm.

2. Supplementation

• Computing the Information Matrix from conditional information via normal approximation (Liu, 1998). The basic idea is to approximate the likelihood function by a normal density when maximum likelihood estimates are assumed to be approximately normally distributed. The method uses two facts: the information for a one-dimensional parameter can be computed when the loglikelihood is approximately quadratic over a range that corresponds to a small positive confidence interval; and the covariance matrix of a normal distribution can be obtained from a set of one-dimensional conditional distributions whose sample spaces span the sample space of the joint distribution.

3. Application

• EM can be used for maximum likelihood estimation of many models, such as multivariate normal, multivariate t, mixed-effects, general location, factor analysis, and mixture models. For example, the EM algorithm has been used in understanding and modelling the relationship among questions/attributes at company level in Customer Value Analysis (CVA). William S. Cleveland and Chuanhai Liu are working on a generalized version of the time series model that has been used as a component in modeling CVA. We implemented the EM algorithm for maximum likelihood estimation of this class of time series models.

• As a supplementary tool for Markov chain Monte Carlo (MCMC) methods for Bayesian computation.

The postscript files of some our current papers are also available.

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