##  [Fisher Information](/fisher-information-0) 

 Definition

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

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

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

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

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

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

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

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

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.