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The E-M algorithm does not guarantee convergence to global maxima. Following Applet shows this nicely:

Is it possible to generate data set, that guarantees reaching global maximum? Maybe few simple constraints imposed on generation process is enough to guarantee, that E-M will find global maximum?

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Increasing the number of data points often helps reducing the number of local maxima. There is no other general recipe to avoid converging to nonglobal maxima with EM, except to ensure that these don't exist.

Or switch the method and use a branch and bound technique. It will find a globab maximum, if it has enough space and time for running to completion. But for typical statistical estimation problems this tends to happen only for fairly small problems.

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