$\Sigma$ norm in Normal Distribution

Question

My professor wrote down this in class: Let $D=N(\mu,\Sigma)$ be a normal distribution, then $$\log(p_D(d))=-\frac{1}{2}||d-\mu||^2_\Sigma+\text{const.}$$

Here I don't completely understand from where the $\Sigma$ came from, as I understand it from wiki it should be $-\frac{1}{2}||d-\mu||^2 \Sigma^{-1}$. Though I am not sure about this.

Thanks for the help.

Answer 1

If it's the norm defined from the Mahalanobis distance (which it seems to be) then the usage is to use the covariance in the notation. The inverse is used in the calculation of the distance.

In this context, we have that $$\|x\|^2_\Sigma=x^T\Sigma^{-1}x$$ where $x$ is a column vector and $\Sigma$ is positive definite.