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Connection between two definitions of Fisher information.

Numerical calculation of fisher informationWhat are the measurement units of Fisher information? (Dimensional Analysis)Reparametrization of multivariable fisher informationShow that Fisher information matrix is the second order gradient of KL divergenceFisher information of sampleIntuition on fisher information on $n$ observations and its relationship with one observationCan Fisher information be zero?Relationship between Fisher information and KL-divergenceFisher Information for a misspecified modelFlatness of a statistical manifold with Fisher information metric

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In statistics, the Fisher information is commonly defined as the covariance matrix$operatornameCov_theta X$ of the random vector $X$, with $X_i = fracpartialpartial theta_i left(log(f(X, theta)right)$. In particular, in the univariate case it boils down to

$$mathbbE_theta left(fracpartialpartial thetalog(f(X, theta))right)^2$$

On the other hand, some sources (usually in context of information theory, such as this one) define the Fisher information of a continuous random variable as

$$mathbbE leftVertnablalog f(X)rightVert^2 = int_mathbbR^n fracleftVertnabla f(x)rightVert^2f(x) dx$$
where $Vert cdot Vert$ denotes the Euclidean norm and $nabla = left(fracpartialpartial x_1, dots, fracpartialpartial x_nright)$

Is there any link between those two definitions apart from their obvious similarity?

statistics information-theory fisher-information

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edited Mar 29 at 7:43

marmistrz

asked Mar 28 at 21:06

marmistrzmarmistrz

616312

$endgroup$

add a comment | 

0

1

$begingroup$

In statistics, the Fisher information is commonly defined as the covariance matrix$operatornameCov_theta X$ of the random vector $X$, with $X_i = fracpartialpartial theta_i left(log(f(X, theta)right)$. In particular, in the univariate case it boils down to

$$mathbbE_theta left(fracpartialpartial thetalog(f(X, theta))right)^2$$

On the other hand, some sources (usually in context of information theory, such as this one) define the Fisher information of a continuous random variable as

$$mathbbE leftVertnablalog f(X)rightVert^2 = int_mathbbR^n fracleftVertnabla f(x)rightVert^2f(x) dx$$
where $Vert cdot Vert$ denotes the Euclidean norm and $nabla = left(fracpartialpartial x_1, dots, fracpartialpartial x_nright)$

Is there any link between those two definitions apart from their obvious similarity?

statistics information-theory fisher-information

share|cite|improve this question

edited Mar 29 at 7:43

marmistrz

asked Mar 28 at 21:06

marmistrzmarmistrz

616312

$endgroup$

add a comment | 

$begingroup$

In statistics, the Fisher information is commonly defined as the covariance matrix$operatornameCov_theta X$ of the random vector $X$, with $X_i = fracpartialpartial theta_i left(log(f(X, theta)right)$. In particular, in the univariate case it boils down to

$$mathbbE_theta left(fracpartialpartial thetalog(f(X, theta))right)^2$$

On the other hand, some sources (usually in context of information theory, such as this one) define the Fisher information of a continuous random variable as

$$mathbbE leftVertnablalog f(X)rightVert^2 = int_mathbbR^n fracleftVertnabla f(x)rightVert^2f(x) dx$$
where $Vert cdot Vert$ denotes the Euclidean norm and $nabla = left(fracpartialpartial x_1, dots, fracpartialpartial x_nright)$

Is there any link between those two definitions apart from their obvious similarity?

statistics information-theory fisher-information

share|cite|improve this question

edited Mar 29 at 7:43

marmistrz

asked Mar 28 at 21:06

marmistrzmarmistrz

616312

$endgroup$

share|cite|improve this question

edited Mar 29 at 7:43

marmistrz

asked Mar 28 at 21:06

marmistrzmarmistrz

616312

share|cite|improve this question

edited Mar 29 at 7:43

marmistrz

asked Mar 28 at 21:06

marmistrzmarmistrz

616312

asked Mar 28 at 21:06

marmistrzmarmistrz

616312

asked Mar 28 at 21:06

marmistrzmarmistrz

616312