We are given the normal density $$p(x) := \frac{1}{\sqrt{(2 \pi)^d \det(\Sigma)}} \exp\left(-\frac{1}{2} (x- \mu)^T \Sigma^{-1} (x-\mu)\right),$$ where we consider the positive definite matrix $\Sigma$ to be fixed.
I want to calculate $\mathbb{E}_{\mu \sim z} [G_{z}G_z^T]$ to find the Fisher kernel, where $$ G_x = \frac{d}{d \mu} \log(p(x)) = \Sigma^{-1} (x - \mu) \log(p(x)). $$ What does $\mathbb{E}_{\mu \sim z}$ mean?
I have seen that notation in machine learning papers to denote the expectation of $\mu$ under the distribution $z$. See for example equation (6) in: https://www.cs.princeton.edu/courses/archive/fall11/cos597C/lectures/variational-inference-i.pdf or page 2 here: https://papers.nips.cc/paper/5423-generative-adversarial-nets.pdf
EDIT: That means that $z$ is a valid distribution for $\mu$.