Can the off-diagonal elements of Fisher information matrix be negative? The concept of Fisher information is new to me and as I understand the diagonal elements of the Fisher information matrix (FIM) are proportional to mean square error (to be precise the inverse of FIM). What is the interpretation of the off-diagonal elements of FIM and can these be negative?

A counter-example: For a Beta $\mathcal B(\alpha,\beta)$ distribution, the Fisher information matrix on $(\alpha,\beta)$ is
$$I(\alpha,\beta) = \left(
\begin{matrix}
\psi'(\alpha) -\psi'(\alpha+\beta) & -\psi'(\alpha+\beta)\\
-\psi'(\alpha+\beta) & \psi'(\beta) -\psi'(\alpha+\beta)
\end{matrix}\right)$$
where $\psi(\cdot)$ is the digamma function and $\psi'(\cdot)>0$.
