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Trying to know the steps to find the maximum likelihood estimate for the covariance matrix, assuming normal probability distribution, I want to differentiate log-likelihood function but what confuses me is that in the equation, shouldn't $l(\mu, \Sigma^{-1})$ be written as $l(\mu, \Sigma)$? How come $ \Sigma $ can be written as $ \Sigma^{-1} $ below in the red circle? For what reason, can I write like that? It is from an online lecture on YouTube.

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What confuses you is the invariance of the maximum likelihood estimators. In particular, for any function $g(\theta)$ of your parameter $\theta$, not necessarily one-to-one, it can be shown that $\widehat{g(\theta)} = g \left(\widehat{\theta} \right)$. Maybe that was mentioned in one of the previous lectures?

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    $\begingroup$ The lecture is the first one. There is no precious one I guess. The lecturer just said that $$|\Sigma|=\frac{1}{|\Sigma^{-1}|}$$ $\endgroup$ – user122358 Mar 2 '17 at 15:18

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