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Entropy 2017, 19(8), 427; doi:10.3390/e19080427

Minimum and Maximum Entropy Distributions for Binary Systems with Known Means and Pairwise Correlations

1
Department of Natural Sciences, Fordham University, New York, NY 10023, USA
2
Department of Physics, University of California, Berkeley, CA 94720, USA
3
Redwood Center for Theoretical Neuroscience, University of California, Berkeley, CA 94720, USA
4
Mathematical Sciences Research Institute, Berkeley, CA 94720, USA
5
Helen Wills Neuroscience Institute, University of California, Berkeley, CA 94720, USA
6
Biophysics Graduate Group, University of California, Berkeley, CA 94720, USA
7
Google Brain, Google, Mountain View, CA 94043, USA
*
Author to whom correspondence should be addressed.
Received: 27 June 2017 / Revised: 8 August 2017 / Accepted: 18 August 2017 / Published: 21 August 2017
(This article belongs to the Special Issue Thermodynamics of Information Processing)
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Abstract

Maximum entropy models are increasingly being used to describe the collective activity of neural populations with measured mean neural activities and pairwise correlations, but the full space of probability distributions consistent with these constraints has not been explored. We provide upper and lower bounds on the entropy for the minimum entropy distribution over arbitrarily large collections of binary units with any fixed set of mean values and pairwise correlations. We also construct specific low-entropy distributions for several relevant cases. Surprisingly, the minimum entropy solution has entropy scaling logarithmically with system size for any set of first- and second-order statistics consistent with arbitrarily large systems. We further demonstrate that some sets of these low-order statistics can only be realized by small systems. Our results show how only small amounts of randomness are needed to mimic low-order statistical properties of highly entropic distributions, and we discuss some applications for engineered and biological information transmission systems. View Full-Text
Keywords: information theory; minimum entropy; maximum entropy; statistical mechanics; Ising model; pairwise correlations; compressed sensing; neural networks information theory; minimum entropy; maximum entropy; statistical mechanics; Ising model; pairwise correlations; compressed sensing; neural networks
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).

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MDPI and ACS Style

Albanna, B.F.; Hillar, C.; Sohl-Dickstein, J.; DeWeese, M.R. Minimum and Maximum Entropy Distributions for Binary Systems with Known Means and Pairwise Correlations. Entropy 2017, 19, 427.

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