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Minimum Mutual Information and Non-Gaussianity Through the Maximum Entropy Method: Theory and Properties
Instituto Dom Luiz, Faculdade de Ciências, University of Lisbon, DEGGE, Ed. C8, Campo-Grande, 1749-016 Lisbon, Portugal
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Received: 20 May 2012; in revised form: 8 June 2012 / Accepted: 15 June 2012 / Published: 19 June 2012
Abstract: The application of the Maximum Entropy (ME) principle leads to a minimum of the Mutual Information (MI), I(X,Y), between random variables X,Y, which is compatible with prescribed joint expectations and given ME marginal distributions. A sequence of sets of joint constraints leads to a hierarchy of lower MI bounds increasingly approaching the true MI. In particular, using standard bivariate Gaussian marginal distributions, it allows for the MI decomposition into two positive terms: the Gaussian MI (Ig), depending upon the Gaussian correlation or the correlation between ‘Gaussianized variables’, and a non‑Gaussian MI (Ing), coinciding with joint negentropy and depending upon nonlinear correlations. Joint moments of a prescribed total order p are bounded within a compact set defined by Schwarz-like inequalities, where Ing grows from zero at the ‘Gaussian manifold’ where moments are those of Gaussian distributions, towards infinity at the set’s boundary where a deterministic relationship holds. Sources of joint non-Gaussianity have been systematized by estimating Ing between the input and output from a nonlinear synthetic channel contaminated by multiplicative and non-Gaussian additive noises for a full range of signal-to-noise ratio (snr) variances. We have studied the effect of varying snr on Ig and Ing under several signal/noise scenarios.
Keywords: mutual information; non-Gaussianity; maximum entropy distributions; non‑Gaussian noise
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Pires, C.A.L.; Perdigão, R.A.P. Minimum Mutual Information and Non-Gaussianity Through the Maximum Entropy Method: Theory and Properties. Entropy 2012, 14, 1103-1126.
Pires CAL, Perdigão RAP. Minimum Mutual Information and Non-Gaussianity Through the Maximum Entropy Method: Theory and Properties. Entropy. 2012; 14(6):1103-1126.
Pires, Carlos A. L.; Perdigão, Rui A. P. 2012. "Minimum Mutual Information and Non-Gaussianity Through the Maximum Entropy Method: Theory and Properties." Entropy 14, no. 6: 1103-1126.