Multivariate gaussian, MLE

[Kiuhnm]

How do you take the derivative of
\(\ln p(X|\mu,\Sigma)=-\frac{N D}{2}\ln(2\pi)-\frac{N}{2}\ln|\Sigma|-\frac{1}{2}\sum_{n=1}^N(x_n-\mu)^T\Sigma^{-1}(x_n-\mu)\)
with respect to \(\Sigma\)?

 

[BGM]

You may first take a look at

http://en.wikipedia.org/wiki/Estima...tion_for_the_multivariate_normal_distribution

 

[Kiuhnm]

I'm really interested in what method you statisticians use when you have to deal with a novel expression (not something you can look up). Do you use matrix calculus? I'm aware of at least 3 approaches:
1) rewrite the expression in non-matrix form, compute the single partials and try to recompact the result into matrix form;
2) use Schonemann's method (paper);
3) use Magnus & Neudecker's method (book).
Just curious.

 

[Dason]

I'm too lazy to follow those links but I just use matrix calculus. The first method you talk about is essentially what matrix calculus will do for you without doing it piece by piece. You just have to learn a few rules of operation and it's really not that bad.

 

[Kiuhnm]

I've just (almost) learned the third method. For a very brief introduction see the first section of this paper, if you're interested.