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Fisher Information Matrix

Let X(x)=X(x_1,x_2,...,x_n) be a random vector in R^n and let f_X(x) be a probability distribution on X with continuous first and second order partial derivatives. The Fisher information matrix of X is the n×n matrix J_X whose (i,j)th entry is given by

(J_X)_(i,j)=<(partiallnf_X(x))/(partialx_i)(partiallnf_X(x))/(partialx_j)>
(1)
=int_(R^n)(partiallnf_X(x))/(partialx_i)(partiallnf_X(x))/(partialx_j)f_X(x)d^nx.
(2)

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Fisher Information

This entry contributed by David Terr

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References

Papathanasiou, V. "Some Characteristic Properties of the Fisher Information Matrix via Cacoullos-Type Inequalities." J. Multivariate Analysis 14, 256-265, 1993.Vignat, C. and Bercher, J.-F. "On Fisher Information Inequalities and Score Functions in Non-Invertible Linear Systems." J. Ineq. Pure Appl. Math. 4, Article 71, 1-9, 2003. http://jipam.vu.edu.au/article.php?sid=312.Zamir, R. "A Proof of the Fisher Information Matrix Inequality Via a Data Processing Argument." IEEE Trans. Information Th. 44, 1246-1250, 1998.Zamir, R. "A Necessary and Sufficient Condition for Equality in the Matrix Fisher Information Inequality." Technical Report, Tel Aviv University, Dept. Elec. Eng. Syst., 1997. http://www.eng.tau.ac.il/~zamir/techreport/crb.ps.gz.

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Fisher Information Matrix

Cite this as:

Terr, David. "Fisher Information Matrix." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/FisherInformationMatrix.html

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