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On Fisher information inequalities and score functions in non-invertible linear systems. (English) Zbl 1056.62005

From the introduction: The Fisher information matrix \(J_X\) of a random vector \(X\) appears as a useful theoretic tool to describe the propagation of information through systems. For instance, it is directly involved in the derivation of the Entropy Power Inequality (EPI), that describes the evolution of the entropy of random vectors submitted to linear transformations. The main contributions of this note are threefold. First, we review some properties of score functions and characterize the estimation of a score function under linear constraints. Second, we give two alternate derivations of R. Zamir’s Fisher information inequalities [IEEE Trans. Inf. Theory 44, 1246-1250 (1998; Zbl 0901.62005)] and show how they can be related to K. Papathanasiou’s results [J. Multivariate Anal. 44, 256–265 (1993; Zbl 0765.62055)]. Third, we examine the cases of equality and give an interpretation that highlights the concepts of extractable components of the input vector of a linear system, and its relationship with the concepts of pseudoinverse and gaussianity.

MSC:

62B10 Statistical aspects of information-theoretic topics
94A17 Measures of information, entropy
93C05 Linear systems in control theory
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