Fisher information

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The amount of information that a sample provides about the value of an unknown parameter. Writing L as the likelihood for n observations from a distribution with parameter θ, Sir Ronald Fisher in 1922 defined the information, I(θ), as being given by . If T is an unbiased estimator of θ then a lower bound to the variance of T is . Since 1948 this has generally been referred to as the Cramér–Rao lower bound in recognition of the independent proofs by Cramér and Rao. The inequality Var(T)≥1/I(θ) (where ‘Var’ denotes variance) is the Cramér–Rao inequality.

If the distribution involves several parameters, θ1, θ2,…, θp, then the Fisher information matrix has elements given by .

Subjects: Probability and Statistics.

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