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单词 FisherInformationMatrix
释义

Fisher information matrix


Given a statistical model {f𝐗(𝒙𝜽)} of a random vector X, the Fisher information matrixMathworldPlanetmath, I, is the varianceMathworldPlanetmath of the score functionMathworldPlanetmath U. So,

I=Var[U].

If there is only one parameter involved, then I is simply called the Fisher information or information of f𝐗(𝒙θ).

Remarks

  • If f𝐗(𝒙𝜽) belongs to the exponential family, I=E[UTU]. Furthermore, with some regularity conditions imposed, we have

    I=-E[U𝜽].
  • As an example, the normal distributionMathworldPlanetmath, N(μ,σ2), belongs to the exponential family and its log-likelihood functionMathworldPlanetmath (𝜽x) is

    -12ln(2πσ2)-(x-μ)22σ2,

    where 𝜽=(μ,σ2). Then the score function U(𝜽) is given by

    (μ,σ2)=(x-μσ2,(x-μ)22σ4-12σ2).

    Taking the derivative with respect to 𝜽, we have

    U𝜽=(U1μU2μU1σ2U2σ2)=(-1σ2-x-μσ4-x-μσ412σ4-(x-μ)2σ6).

    Therefore, the Fisher information matrix I is

    -E[U𝜽]=12σ4(2σ200-1).
  • Now, in linear regression model with constant variance σ2, it can be shown that the Fisher information matrix I is

    1σ2𝐗T𝐗,

    where X is the design matrix of the regression model.

  • In general, the Fisher information meansures how much “information” is known about a parameter θ. If T is an unbiased estimatorMathworldPlanetmath of θ, it can be shown that

    Var[T(X)]1I(θ)

    This is known as the Cramer-Rao inequality, and the number 1/I(θ) is known as the Cramer-Rao lower bound. The smaller the variance of the estimate of θ, the more information we have on θ. If there is more than one parameter, the above can be generalized by saying that

    Var[T(X)]-I(𝜽)-1

    is positive semidefinite, where I is the Fisher information matrix.

TitleFisher information matrix
Canonical nameFisherInformationMatrix
Date of creation2013-03-22 14:30:15
Last modified on2013-03-22 14:30:15
OwnerCWoo (3771)
Last modified byCWoo (3771)
Numerical id14
AuthorCWoo (3771)
Entry typeDefinition
Classificationmsc 62H99
Classificationmsc 62B10
Classificationmsc 62A01
Synonyminformation matrix
DefinesFisher information
Definesinformation
DefinesCramer-Rao inequality
DefinesCramer-Rao lower bound
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