Scorefunktion

Score function is a concept in statistics that refers to the gradient of the log-likelihood with respect to model parameters. For a parametric model with data X1, …, Xn and joint density f(x; θ), the log-likelihood is ℓ(θ) = ∑ log f(Xi; θ). The score vector is U(θ) = ∂ℓ(θ)/∂θ, a random vector that summarizes how the likelihood responds to small changes in θ.

A fundamental property is that, at the true parameter θ0, the expected score vanishes: Eθ0[U(θ0)] = 0. The variability of the score is captured by the Fisher information I(θ0) = Varθ0[U(θ0)] = −Eθ0[∂²ℓ(θ)/∂θ∂θᵀ]. These relationships underpin the asymptotic theory of maximum likelihood estimation: under regularity conditions, the maximum likelihood estimator θ̂ solves U(θ̂) = 0 and is consistent and asymptotically normal with covariance I(θ0)⁻¹/n.

The score also underlies hypothesis testing. The score test (Lagrange Multiplier test) uses U(θ0) and I(θ0) to assess H0: θ = θ0, with a test statistic that, under H0, follows a χ² distribution with degrees of freedom equal to the number of tested parameters.

In addition to likelihood-based interpretations, the term score can refer to score matching, a method for estimating unnormalized densities by matching the score functions sθ(x) = ∇ log pθ(x) of the model to the data, minimizing the expected squared difference.

Example: for a one-parameter normal model with known variance, the score for μ is U(μ) = ∑(Xi − μ)/σ², and setting U(μ) = 0 yields the sample mean as the MLE.

Related concepts include Fisher information, likelihood, and estimating equations.