Minimum and maximum entropy distributions for binary systems with known means and pairwise correlations

TL;DR

This paper derives bounds and constructs low-entropy distributions for binary systems with fixed means and pairwise correlations, revealing how small randomness can replicate complex statistical properties.

Contribution

It provides the first bounds on entropy for distributions with fixed means and pairwise correlations and constructs explicit low-entropy solutions for large systems.

Findings

Minimum entropy scales logarithmically with system size.

Some low-order statistics are only realizable in small systems.

Small randomness suffices to mimic complex statistical properties.

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 {\em 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.