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DEFINITION

The Fisher information is the amount of Information that an observable Random Variable ''X'' carries about an unknown Parameter θ upon which the Likelihood Function of heta , L( heta)= f(X; heta), depends. The likelihood function is the joint probability of the data, the ''X''s, conditional on the value of θ, ''as a function of θ''. Since the Expectation of the Score is zero, the Variance is simply the second Moment of the score, the derivative of the Log of the Likelihood Function with respect to θ. Hence the Fisher information can be written

:
\mathcal{I}( heta)
=
\mathrm{E}
\left\{\left.
\left[
rac{\partial}{\partial heta} \ln f(X; heta)
ight]^2


rac{\partial \Sigma_{1,2}}{\partial heta_m} &
\cdots &
rac{\partial \Sigma_{1,N}}{\partial heta_m} \ \
rac{\partial \Sigma_{2,1}}{\partial heta_m} &
rac{\partial \Sigma_{2,2}}{\partial heta_m} &
\cdots &
rac{\partial \Sigma_{2,N}}{\partial heta_m} \ \
dots & dots & \ddots & dots \ \
rac{\partial \Sigma_{N,1}}{\partial heta_m} &
rac{\partial \Sigma_{N,2}}{\partial heta_m} &
\cdots &
rac{\partial \Sigma_{N,N}}{\partial heta_m}
\end{bmatrix}.



SEE ALSO


Other measures employed in Information Theory :


REFERENCES