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36, 239–249 (1984) 9. V. Delaubert, N. Treps, C. Fabre, A. A. Bachor, P. Réfrégier, Quantum limits in image processing. Europhys. Lett. 81, 44001 (2008) 10. A. Dubois, K. Grieve, G. Moneron, R. Lecaque, L. Vabre, C. Boccara, Ultrahigh-resolution full-field optical coherence tomography. Appl. Opt. 43(14), 2874–2883 (2004) References 45 11. A. Fisher, On the mathematical foundations of theoretical statistics. Philos. Trans. Roy. Soc. Lond. 222, 309–368 (1922) 12. A. Fisher, Theory of statistical estimation.

IEEE Trans. Signal Process. 42, 2859 (1994) 29. W. Weaver, Some Recent Contributions to the Mathematical Theory of Communication (University of Illinois Press, Urbana, 1949) 30. N. Wiener, Generalized harmonic analysis. Acta Math. 55, 117–258 (1930) Chapter 3 Information and Estimation Theory A likely impossibility is always preferable to an unconvincing possibility. 1 Introduction Shannon’s seminal paper of 1948 [32] is popularly seen as the cornerstone of information theory. Arguably Shannon’s paper addressed a significant problem of the age, namely the scientific quantification of the rather abstract concept of information.

Furthermore, Eq. 4) in conjunction with Eq. 25) provides a second definition for the FIM ⎤ ⎡ † † † ∂sw ∂ ∂ ⎦. 26) = −E ⎣ Jw = −E ∂w ∂w ∂w The second order derivative found in the kernel of this expectation can be interpreted as a measure of the curvature of the PDF as the parameter value w is varied. If this curvature is large this again equates to a strong dependence of the observed data X on w. Biased estimators, however, present a further complication because they render Eq. 24) invalid. Bias can, for example be introduced if the parameters w are a non-linear function of the measured data.

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