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The empirical score outer-product matrix isUnder regularity and consistency of $\hat{\theta }$, $i_n(\hat{\theta })$ converges in probability to the Fisher information matrix. This convention differs from the negative-Hessian convention for Observed Fisher information.

For $X\sim N_p(\theta ,\Sigma )$ with known nonsingular covariance matrix, the score function is ${\Sigma }^{-1}(X-\theta )$ and the per-observation Fisher information matrix is ${\Sigma }^{-1}$. In particular, covariance $I_p$ gives information $I_p$.

The Jeffreys prior uses the square root of the determinant of the Fisher information matrix as a parameter-density kernel. It is invariant under smooth one-to-one reparameterization, since the information and density Jacobians transform compatibly. It may be an improper prior; invariance does not guarantee posterior propriety. With nuisance parameters, a scalar conditional Jeffreys prior and the joint Jeffreys prior need not coincide.

The observed Fisher information is the negative Hessian of the realized log-likelihood. Its expectation, under standard regularity conditions, is the Fisher information matrix.