I am given that the second derivative of the log-likelihood to be $$ \frac{d^2\ell}{d\boldsymbol{\mu}^2} = -\Sigma^{-1}*n $$

$\Sigma$ is the covariance matrix and $n$ is the number of random variables. Why is this expression always negative? Is there some property of the covariance matrix that I am unaware of that makes the inverse of this matrix only have positive elements?

  • $\begingroup$ @Winther I don't necessarily think that, but I understand that the Gaussian is a strictly concave distribution with respect to the mean vector $\boldsymbol{\mu}$. So wouldn't that necessitate that the second derivative of the log-likelihood be always negative? $\endgroup$ – Iamanon Oct 4 '18 at 12:56
  • $\begingroup$ But the right hand side is not a number, it's a matrix. By positive/negative do you mean it's positive/negative definite or something else? $\endgroup$ – Winther Oct 4 '18 at 13:01
  • $\begingroup$ No. So perhaps I'm thinking about this the wrong way. But in scalar functions, if we find that it's second derivative is always negative, we can conclude that the function is strictly concave. I am trying to apply that same idea here to show that $\ell$ is always concave, so I thought its second derivative should always be negative? But as you said, it is a matrix so I guess I don't understand how this works. $\endgroup$ – Iamanon Oct 4 '18 at 13:07
  • $\begingroup$ Why do need to show it's concave (whatever that means here)? What is the underlying problem you want to solve or what is the thing you are trying to understand? I suggest asking directly about that. $\endgroup$ – Winther Oct 4 '18 at 13:14

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