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Relations Between Information and Estimation in the Presence of Feedback

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Part of the Lecture Notes in Control and Information Sciences book series (LNCIS,volume 450)

Abstract

We discuss some of the recent literature on relations between information- and estimation-theoretic quantities. We begin by exploring the connections between mutual information and causal/non-causal, matched/mismatched estimation for the setting of a continuous-time source corrupted by white Gaussian noise. Relations involving causal estimation, in both matched and mismatched cases, and mutual information persist in the presence of feedback. We present a new unified framework, based on Girsanov theory and Itô’s Calculus, to derive these relations. We conclude by deriving some new results using this framework.

Keywords

  • Mutual Information
  • Relative Entropy
  • Estimation Loss
  • Standard Brownian Motion
  • Gaussian Channel

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Notes

  1. 1.

    Define the filtration \(\mathcal{F}^{Y}_{t} = \sigma\{ Y(B) : B \subseteq\{s: s < t\} \}\). Note that in the setting of Theorem 5.5, the encoder ϕ t is measurable w.r.t. the σ-algebra \(\mathcal{F}^{X}_{T} \vee\mathcal{F}^{Y}_{t}\), and the estimate \(\hat{\phi}_{t}\) is measurable w.r.t. (or adapted to the filtration) \(\mathcal{F}^{Y}_{t}\).

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Acknowledgement

This research was supported by LCCC—Linnaeus Grant VR 2007-8646, Swedish Research Council.

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Correspondence to Himanshu Asnani .

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Asnani, H., Venkat, K., Weissman, T. (2014). Relations Between Information and Estimation in the Presence of Feedback. In: Como, G., Bernhardsson, B., Rantzer, A. (eds) Information and Control in Networks. Lecture Notes in Control and Information Sciences, vol 450. Springer, Cham. https://doi.org/10.1007/978-3-319-02150-8_5

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  • DOI: https://doi.org/10.1007/978-3-319-02150-8_5

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-02149-2

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