A nonnegative quantity associated with a parametric statistical model, defined as the expected value of the square of the score (derivative of log-likelihood) or equivalently the negative expected second derivative of the log-likelihood; it measures local sensitivity of the likelihood to parameter changes.

Principle

Quantify how much an observable random variable reveals about a parameter by measuring the curvature of the log-likelihood: larger Fisher information implies greater precision attainable for unbiased estimators.

Demonstration

For independent samples X1,...,Xn ~ N(μ,σ^2) with known σ^2, the Fisher information about μ is n/σ^2, reflecting that variance of the sample mean decreases proportionally to 1/n.

Misapplication

Interpreting Fisher information as directly equal to an estimator's variance without accounting for model misspecification, bias, or failure of regularity conditions required to apply the Cramér–Rao bound.

Consequence

Leads to the Cramér–Rao lower bound limiting the variance of unbiased estimators and appears in asymptotic normality of maximum-likelihood estimators where the inverse information gives the asymptotic covariance.

Reversal

Zero Fisher information for a parameter component indicates the data carry no local information about that parameter (likelihood flat in that direction); reversing the notion yields parameter non-identifiability rather than high precision.

Boundary

Defined when the likelihood is differentiable and the score's square is integrable under the model; not applicable for models with singular measures, nondifferentiable likelihoods, or parameters on the boundary where regularity fails.

Semantic Tension

Often contrasted with Shannon information: Fisher information measures local parameter sensitivity in a statistical model, whereas Shannon information measures uncertainty reduction in a random variable's outcomes.

Synthesis

Fisher information is the expected curvature of the log-likelihood with respect to parameters, quantifying how strongly data constrain local parameter perturbations and determining asymptotic estimator precision.