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While reading on Gaussian Discriminant Analysis, I came across a derivation of the parameters (specifically, $\sum$) using Maximum Likelihood Estimation that claimed that the MLE of $\sum$ is equivalent to the MLE of $\sum^{-1}$, $\sum$ being the covariance matrix of the multivariate Gaussian distribution.

Though I am able to reach the same result without using the above, I want to know:

  1. Why does this hold true?
  2. Does this hold true for any positive definite matrix $A \in \mathbb S_{++}^{n}$, or does the fact that $\sum$ is a covariance matrix play a role in proving this.
CptFoobar
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  • See https://stats.stackexchange.com/questions/80584/invariance-property-of-maximum-likelihood-estimator – kjetil b halvorsen Jul 27 '20 at 19:36
  • @kjetilbhalvorsen that helps to answer my questions. Would you like to formulate that into a point-wise answer for anyone else that may run into this? – CptFoobar Aug 04 '20 at 11:54

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