Weighted maximum likelihood estimator for poisson variables

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I have been learning about the E-M algorithm in the Normal case. I would like to create the algorithm with two Poisson variables. To do so, I need to calculate the weighted MLE as seen here in the M-Step. I have tried a number of techniques but my code shows I am doing this completly wrong. Any ideas?

Ideally I can recreate this "M-Step" for my setting by calculating this MLE.