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?

TL;DR: The information matrix can be diagonalized, in which case its diagonal elements are positive, whereas non-diagonal elements are zero.
A meaningful information matrix must be positive-definite, but there is no requirements that non-diagonal elements must be positive. That such elements are present indicates that the chosen variables are correlated, and one should exercise care when interpreting diagonal elements as mean square error. This interpretation is correct, if we diagonalize the information matrix, in which case the diagonal elements are positive, while non-diagonal elements are zero.
