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I am following along in Bishop's Pattern Recognition and ML chapters 9 and 10, and I understand that the EM algorithm works by iteratively updating model parameters using equations derived from setting the derivative of the log likelihood to zero - i.e. maximum likelihood estimation.

But for doing GMMs by variational inference, you are finding the model parameters by maximising the evidence lower bound and driving the approximate q() function towards the posterior. I know it is useful to know the posterior, but how exactly does maximising the evidence LB give us a best fit model of the data? Is this maximum likelihood estimation, or MAP estimation?

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