Multivariate normal distribution [edit] The FIM for a $N$-variate multivariate normal distribution, $X \sim N(\mu(\theta), \Sigma(\theta))$ has a special form. Let the $K$-dimensional vector of parameters be $\theta=\left[\begin{array}{lll}\theta_{1} & \ldots & \theta_{K}\end{array}\right]^{\top}$ and the vector of random normal variables be $X=\left[\begin{array}{lll}X_{1} & \ldots & X_{N}\end{array}\right]^{\top} .$ Assume that the mean values of these random variables are $\mu(\theta)=\left[\mu_{1}(\theta) \quad \ldots \quad \mu_{N}(\theta)\right]^{\top}$, and let $\Sigma(\theta)$ be the covariance matrix. Then, for $1 \leq m, n \leq K$, the $(m, n)$ entry of the FIM is: $^{[16]}$ $$ \mathcal{I}_{m, n}=\frac{\partial \mu^{\top}}{\partial \theta_{m}} \Sigma^{-1} \frac{\partial \mu}{\partial \theta_{n}}+\frac{1}{2} \operatorname{tr}\left(\Sigma^{-1} \frac{\partial \Sigma}{\partial \theta_{m}} \Sigma^{-1} \frac{\partial \Sigma}{\partial \theta_{n}}\right) $$ where $(\cdot)^{\top}$ denotes the transpose of a vector, $\operatorname{tr}(\cdot)$ denotes the trace of a square matrix, and: $$ \begin{aligned} \frac{\partial \mu}{\partial \theta_{m}} &=\left[\begin{array}{lll} \frac{\partial \mu_{1}}{\partial \theta_{m}} & \frac{\partial \mu_{2}}{\partial \theta_{m}} & \cdots & \frac{\partial \mu_{N}}{\partial \theta_{m}} \end{array}\right]^{\top} \\ \frac{\partial \Sigma}{\partial \theta_{m}} &=\left[\begin{array}{cccc} \frac{\partial \Sigma_{1,1}}{\partial \theta_{m}} & \frac{\partial \Sigma_{1,2}}{\partial \theta_{m}} & \cdots & \frac{\partial \Sigma_{1, N}}{\partial \theta_{m}} \\ \frac{\partial \Sigma_{2,1}}{\partial \theta_{m}} & \frac{\partial \Sigma_{2,2}}{\partial \theta_{m}} & \cdots & \frac{\partial \Sigma_{2, N}}{\partial \theta_{m}} \\ \vdots & \vdots & \ddots & \vdots \\ \frac{\partial \Sigma_{N, 1}}{\partial \theta_{m}} & \frac{\partial \Sigma_{N, 2}}{\partial \theta_{m}} & \cdots & \frac{\partial \Sigma_{N, N}}{\partial \theta_{m}} \end{array}\right] \end{aligned} $$
Most importantly: What is the variable "m" in the definition of the multivariate normal Fisher Information??
This Wikipedia Definition does not make sense as the fisher information is a metric tensor induced by the hypothesis space and therefore is guaranteed to be symmetric. Writting each entry of the fisher information as $\mathcal{I}_{m,n}$ where $1\leq m, n\leq K$ suggests the Fisher Information Matrix is not symmetric. We do not even have an upper bound on "m"!
Also Is K, the # parameters equal to the N, the number of different mean variables?
What's supposed to be the dimensions of the Fisher info for a multivariate normal?
