Invariance of the Fisher matrix

In summary, the Fisher matrix is the average of the second derivative of the log-likelihood with respect to the parameters. It is invariant to any non-singular linear transformation of the data, which is a useful property. This is confirmed by the fact that the Fisher information matrix remains unchanged under a transformation of the data. The Biometrika (1998) article provides further discussion on this topic.
  • #1
lukluk
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Fisher matrix=(minus the) average of the second derivative of the log-likelihood with respect to the parameters

It seems to me the Fisher matrix for Gaussian data is invariant with respect to any (non-singular) linear transformation of the data; if correct this is a very useful property, however I cannot find a reference to this in the texts available to me. Can anyone confirm whether this is true or refer me to some discussion of this?

Thanks!
 
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  • #2
lukluk said:
Fisher matrix=(minus the) average of the second derivative of the log-likelihood with respect to the parameters

It seems to me the Fisher matrix for Gaussian data is invariant with respect to any (non-singular) linear transformation of the data; if correct this is a very useful property, however I cannot find a reference to this in the texts available to me. Can anyone confirm whether this is true or refer me to some discussion of this?

Thanks!

The Fisher information matrix is equivalent to the reciprocal of the asymptotic variance-covariance matrix of the parameter. Under the transformation [itex]T(\vec{x})=A(\vec{x})[/itex] the information content of the matrix is unchanged.

Biometrika(1998)85,4 pp973-979
 
Last edited:
  • #3
great thanks!
 

What is the Fisher matrix?

The Fisher matrix is a mathematical tool used in statistics and data analysis, specifically in the field of parameter estimation. It is used to calculate the variances and covariances of the parameters of a statistical model.

What is the importance of invariance in the Fisher matrix?

Invariance in the Fisher matrix refers to the property that the matrix remains the same even when the parameters of the statistical model are transformed. This is important because it allows for the comparison of different models and the calculation of confidence intervals for the parameters.

How is invariance achieved in the Fisher matrix?

Invariance in the Fisher matrix is achieved by using a transformation matrix to change the parameters of the model, while keeping the same underlying statistical model. This ensures that the resulting Fisher matrix is the same, regardless of the parameter transformation.

Can the Fisher matrix be used for any statistical model?

Yes, the Fisher matrix can be used for any statistical model as long as it is differentiable and has a finite number of parameters. It is commonly used in fields such as physics, engineering, and economics to estimate the parameters of complex models.

How is the invariance of the Fisher matrix tested?

The invariance of the Fisher matrix can be tested by transforming the parameters of the model and comparing the resulting Fisher matrices. If the matrices are the same, then the Fisher matrix is invariant. Additionally, theoretical proofs can also be used to demonstrate the invariance property.

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