Higher-order Common Information

Abstract

Shannon's mutual information quantifies redundancy between two random variables. We introduce a new notion, termed higher-order common information (HCI), which captures the information shared among $n$ arbitrarily distributed random variables. The quantity is defined through an iterative information-bottleneck construction and can be interpreted as the maximum rate at which a single compressed representation can simultaneously preserve information about all variables. For jointly Gaussian and Bernoulli sources, we derive closed-form expressions for any $n$. We furthermore show that the HCI yields strictly tighter characterizations of redundancy than existing bounds, and demonstrate how to numerically approximate the HCI for arbitrarily distributed sources.

This is a significant rewrite. Irrelevant sections are removed and the storyline rewritten. The higher-order common information is formerly defined, examples provided, and corresponding closed-form expressions for Gaussian and Bernoulli sources are found for any number of variables