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The variance of a MLE estimate can increase or decrease depending on the samples.

In maximum likelihood estimate mle, we find a gaussian that fits the data in a certain way.

I was under the impression that MAP and MLE are same. Probably because they are taught as limiting cases. But they have entirely different formulation. For ex. MLE variance can increase or decrease whereas MAP variance can only ever decrease.

yellow circles + lines are samples $x_i$, red circle, green line is the mle $(\mu, \sigma)$