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On mutual information, likelihood ratios, and estimation error for the additive Gaussian channel

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1 Author(s)
M. Zakai ; Dept. of Electr. Eng., Technion-Israel Inst. of Technol., Haifa, Israel

This paper considers the model of an arbitrarily distributed signal x observed through an added independent white Gaussian noise w, y=x+w. New relations between the minimal mean-square error of the noncausal estimator and the likelihood ratio between y and w are derived. This is followed by an extended version of a recently derived relation between the mutual information I(x;y) and the minimal mean-square error. These results are applied to derive infinite-dimensional versions of the Fisher information and the de Bruijn identity. A comparison between the causal and noncausal estimation errors yields a restricted form of the logarithmic Sobolev inequality. The derivation of the results is based on the Malliavin calculus

Published in:

IEEE Transactions on Information Theory  (Volume:51 ,  Issue: 9 )