Maximum Likelihood Estimation via the ECM Algorithm: A General Framework
Xiao‐Li Menga b, Donald B. Rubinc d
- aDepartment of Statistics, University of Chicago Chicago, Illinois 60637, U.S.A
- bDepartment of Statistics, University of ChicagoChicago, Illinois 60637, U.S.A
- cDepartment of Statistics, Harvard University Cambridge, Massachusetts 02138, U.S.A
- dDepartment of Statistics, Harvard UniversityCambridge, Massachusetts 02138, U.S.A
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Abstract
Two major reasons for the popularity of the EM algorithm are that its maximum step involves only complete-data maximum likelihood estimation, which is often computationally simple, and that its convergence is stable, with each iteration increasing the likelihood. When the associated complete-data maximum likelihood estimation itself is complicated, EM is less attractive because the M-step is computationally unattractive. In many cases, however, complete-data maximum likelihood estimation is relatively simple when conditional on some function of the parameters being estimated. We introduce a class of generalized EM algorithms, which we call the ECM algorithm, for Expectation/Conditional Maximization (CM), that takes advantage of the simplicity of complete-data conditional maximum likelihood estimation by replacing a complicated M-step of EM with several computationally simpler CM-steps. We show that the ECM algorithm shares all the appealing convergence properties of EM, such as always increasing the likelihood, and present several illustrative examples.