A symmetric positive semidefinite matrix whose (i,j) element equals the expected value of the product of score components or the negative expected second derivative of the log-likelihood with respect to parameters; it quantifies parameter sensitivity of a parametric model.

Principle

Organizes local parameter identifiability and sensitivity: information adds across independent observations and governs lower bounds on estimator variance.

Demonstration

For n independent observations from N(μ,σ^2) with known σ, the Fisher information for μ is n/σ^2; for a vector parameter the matrix entries follow similarly from expected score products.

Misapplication

Treating the Fisher information as reliable in small samples without checking regularity, using it when the model is misspecified, or inverting a singular matrix as if it were nonsingular.

Consequence

When regularity holds, the inverse Fisher matrix gives the asymptotic covariance of the maximum likelihood estimator and determines the Cramér–Rao lower bound for unbiased estimators.

Reversal

The observed information replaces expectation by the actual negative Hessian at the data; in the limit of vanishing information the inverse diverges, indicating non-identifiability.

Boundary

Applies to parametric models with differentiable likelihoods and finite expectations of score products; excludes infinite-dimensional or non-differentiable parameterizations and cases with infinite second moments.

Semantic Tension

Sometimes conflated with the observed (sample) negative Hessian; the two coincide in expectation but differ numerically, especially in small samples or misspecified models.

Synthesis

The Fisher information matrix is the expectation-based, additive measure of how much a parametric model's likelihood reveals about its parameters, governing asymptotic estimator variability.