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I need to calculate the $\dfrac{\partial \log |\Sigma|}{\partial \rho }$ when $\Sigma = (1-\rho) I + \rho \mathbf{1} \mathbf{1}^\top$ and $\Sigma$ has dimension $p \times p$.

I try to use the formula presented here and here but the result is not right.

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  • $\begingroup$ Without trying anything fancy, you can simply calculate $|\Sigma|$ and proceed as usual. $\endgroup$ – StubbornAtom Apr 25 at 15:38
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The derivative wrt $\rho$ is $\operatorname{trace}((-I+11^T){\Sigma}^{-1})$.

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