FisherInformationsfunktion

Fisher information, named after Ronald A. Fisher, is a measure of the amount of information that an observable random variable X carries about an unknown parameter θ of a statistical model.

For a parametric family with density f(x; θ), the score is U(θ) = ∂/∂θ log f(X; θ). The Fisher information is I(θ) = E_θ[U(θ)^2] = -E_θ[∂^2/∂θ^2 log f(X; θ)]. This quantity captures how much the likelihood concentrates around the true parameter value as data are observed.

For a vector parameter θ ∈ R^k, the information is the k×k Fisher information matrix I(θ) with entries I_{ij}(θ) = E_θ[ ∂ log f(X; θ)/∂θ_i · ∂ log f(X; θ)/∂θ_j ].

For independent and identically distributed samples X1,...,Xn, the total information is I_n(θ) = n I_1(θ). This additive property reflects the cumulative information provided by multiple observations.

Common examples include Bernoulli(p) with I(p) = n/[p(1−p)], and a normal model with known variance σ^2 where I(μ) = n/σ^2. These illustrate how information depends on the underlying distribution and parameters.

A central result is the Cramér-Rao bound: Var(θ̂) ≥ I(θ)^{-1} for unbiased estimators. In the multivariate case, Cov(θ̂) ≥ I(θ)^{-1}. Fisher information is additive for independent trials, invariant under reparameterization, and governs the asymptotic efficiency of maximum likelihood estimators. It also underpins Jeffreys priors via sqrt(det I(θ)).

Historically, Fisher introduced the concept in the 1920s as part of estimation theory. It remains a foundational concept in statistical inference, experimental design, and information theory.

See also: likelihood, score, Cramér–Rao bound, asymptotic normality.