# In Fisher information matrix, how are estimators of variances given?

Can I use the Fisher information matrix to derive estimators for variances of unknown parameters?

I know that for the Fisher information (non-matrix form) the variance of, say, $\theta$ is given by the inverse of $J(\theta)$ (the Fisher information function).

But what about for multiple (e.g. 2) variables and the Fisher information matrix form?

• In case the fisher information $J(\theta)$ is a matrix the size $n \times n$ with $n > 1$ the variance of the parameters are still given by the inverse of the fisher information. i.e. $J(\theta)^{-1}$. However, inverting a matrix is slightly more tricky than inverting a scalar. You need to find the matrix $B$ whose matrix-product with $J(\theta)$ results in the identity matrix $I$. Feb 24, 2016 at 10:17

The variances of the parameters (of a distribution that's compatible with the requirements of Fisher information) are found from the diagonal of the inverse $J(\theta)^{-1}$ of the Fisher information matrix.
• "Variables of a distribution" isn't a standard term: if you mean "parameters of a distribution", as suggested by the $\theta$ notation in your question, then these are unknown constants & don't have a variance. Fisher information relates to the variance of estimators, but how? & to which estimators? Feb 24, 2016 at 13:27